foundations

**[BOOK]**** ****A
mathematical theory of evidence**

G Shafer – Princeton University Press, 1976

Both in science and in practical
affairs we reason by combining facts only inconclusively supported by evidence.
Building on an abstract understanding of this process of combination, this book
constructs a new theory of epistemic probability. The theory draws on the work
of A. P. Dempster but diverges from Depster's viewpoint by identifying his
"lower probabilities" as epistemic probabilities and taking his rule
for combining "upper and lower probabilities" as fundamental.

The book opens with a critique of the
well-known Bayesian theory of epistemic probability. It then proceeds to
develop an alternative to the additive set functions and the rule of
conditioning of the Bayesian theory: set functions that need only be what
Choquet called "monotone of order of infinity." and Dempster's rule
for combining such set functions. This rule, together with the idea of
"weights of evidence," leads to both an extensive new theory and a
better understanding of the Bayesian theory. The book concludes with a brief
treatment of statistical inference and a discussion of the limitations of
epistemic probability. Appendices contain mathematical proofs, which are
relatively elementary and seldom depend on mathematics more advanced that the
binomial theorem.

Cited by 13__532__

**+ [book] ****Advances in the
Dempster-Shafer theory of evidence**

R Yager, M Fedrizzi, J Kacprzyk (editors) – John Wiley, 1994

http://www.amazon.com/Advances-Dempster-Shafer-Theory-Evidence-Ronald/dp/0471552488

It is with great
pleasure that I welcome this collection of diverse and stimulating
contributions to the Dempster-Shafer theory of belief functions. These
contributions demonstrate the vigor and fruitfulness of current research on
belief functions, and their publication as a unit can serve to make that
research even more vigorous. During the past decade, research on belief
functions has suffered from fragmentation; the researchers involved have been
spread over so many different disciplines, meetings, and journals that they
have often been unaware of each other's work. By bringing together so many of
the leading workers in the field, the editors of this volume have begun to
create a new research community. Though belief-function ideas can be found in
the eighteenth-century literature on the probability of testimony, the modern
theory has its origins in work by A.P. Dempster in the 1960s. Dempster was
inspired by R.A. Fisher's “fiducial” method—a brilliant but unsatisfactory
method for computing probabilities for statistical parameters from
observations. Dempster's generalization of Fisher's method produced
non-additive probabilities, which combined by a general rule that I later
called “Dempster's rule of combination.”

debate

+ **Review of a mathematical theory
of evidence**

LA Zadeh - AI magazine, Vol.5, No. 3, 1984 - DOI: http://dx.doi.org/10.1609/aimag.v5i3.452

The seminal work of Glenn
Shafer-which is based on an earlier work of Arthur Dempster-was published at a
time when the theory of expert systems was in its infancy and there was little
interest within the AI community in issues relating to probabilistic or
evidential reasoning. Recognition of the relevance of the Dempster-Shafer
theory to the management of uncertainty in expert systems was slow in coming.
Today, it is the center of considerable attention within AI due in large
measure to (a) the emergence of expert systems as one of the most significant
areas of activity in knowledge engineering, and (b) the important extensions,
applications and implementations of Shafer’s theory made by John Lowrance at
SRI International, Jeff Barnett at USC/ISI, and Ted Shortliffe and Jean Gordon
at Stanford University. What are the basic ideas behind the Dempster-Shafer
theory? In what ways is it relevant to expert systems? What are its
potentialities and limitations? My review of Shafer’s book will be more of an
attempt to provide some answers to these and related questions than a chapter-by-chapter
analysis of its contents.

debate

LA Zadeh - AI magazine, Vol. 7, No. 2, 1986 - DOI: http://dx.doi.org/10.1609/aimag.v7i2.542

During the past two years, the Dempster-Shafer
theory of evidence has attracted considerable attention within the AI community
as a promising method of dealing with uncertainty in expert systems. As
presented in the literature, the theory is hard to master. In a simple approach
that is outlined in this paper, the Dempster-Shafer theory is viewed in the
context of relational databases as the application of familiar retrieval
techniques to second-order relations in first normal form. The relational
viewpoint clarifies some of the controversial issues in the Dempster-Shafer
theory and facilities its use in AI-oriented applications.

frameworks

**A mathematical theory of hints (an approach to the Dempster-Shafer theory of evidence)**

J Kohlas, PA Monney - Lecture Notes in Economics and Mathematical Systems, Vol.
425 -
ISSN 0075-8442

https://www.researchgate.net/publication/239295007_A_Mathematical_Theory_of_Hints_An_Approach_to_the_Dempster-Shafer_Theory_of_Evidence

The subject of the book is an approach to the modeling of
and the reasoning under uncertainty. It develops the Dempster-Shafer Theory as
a theory of the reliability of reasoning with uncertain arguments. A particular
interest of this approach is that it yields a new synthesis and integration of
logic and probability theory. The reader will benefit from a new view at
uncertainty modeling which extends classical probability theory.

TABLE
OF CONTENTS

1.
Introductory Examples.- 2. The Mathematical Concept of a Hint.- 3. Support,
Credibility, Plausibility and Possibility.- 4. Combining Hints.- 5.
Probabilistic Assumption-Based Reasoning.- 6. Rule-Based Systems With
Unreliable Rules.- 7. Compatible Frames of Discernment.- 8. Reasoning on
Compatible Frames.- 9. Statistical Inference.- 10. Describing Uncertainty in
Dynamical Systems.- 11. Diagnostics.- 12. Temporal and Spatial Reasoning.- 13.
The General Model of a Hint.- 14. Structure of Support and Plausibility.- 15.
Dempster’s Rule in the General Case.- 16. Closed Random Intervals.- References.

computation

+ **Computational
methods for a mathematical theory
of evidence**

JA Barnett - Classic
Works of the Dempster-Shafer Theory of Belief Functions, Studies in Fuzziness and Soft
Computing Volume 219, 2008, pp
197-216

Many knowledge-based expert systems employ numerical schemes to
represent evidence, rate competing hypotheses, and guide search through the
domain’s problem space. This paper has two objectives: first, to introduce one
such scheme, developed by Arthur Dempster and Glen Shafer, to a wider audience;
second, to present results that can reduce the computation-time complexity from
exponential to linear, allowing this scheme to be implemented in many more
systems. In order to enjoy this reduction, some assumptions about the structure
of the type of evidence represented and combined must be made. The assumption
made here is that each piece of the evidence either confirms or denies a single
proposition rather than a disjunction. For any domain in which the assumption
is justified, the savings are available.

computation

**Approximations for efficient computation in the theory of evidence**

B Tessem - Artificial Intelligence 61, Issue 2, June 1993, Pages 315–329

http://www.sciencedirect.com/science/article/pii/000437029390072J

The theory of evidence has become a widely used method
for handling uncertainty in intelligent systems. The method has, however, an
efficiency problem. To solve this problem there is a need for approximations.
In this paper an approximation method in the theory of evidence is presented.
Further, it is compared experimentally with Bayesian and consonant
approximation methods with regard to the error they make. Depending on
parameters and the nature of evidence the experiments show that the new method
gives comparatively good results. Properties of the approximation methods for
presentation purposes are also discussed.

advances – entropy - information

+ **Entropy and specificity in a mathematical theory of evidence**

RR Yager - International Journal of General System 9, 249-260, 1983

http://www.tandfonline.com/doi/abs/10.1080/03081078308960825?journalCode=ggen20

We review Shafer's theory of evidence. We then introduce the
concepts of entropy and specificity in the framework of Shafer's theory. These
become complementary aspects in the indication of the quality of evidence.

computation - approximation

M Bauer - International Journal of Approximate Reasoning 17, Issues 2–3, August–October 1997, 217–237

The computational complexity of reasoning within
the Dempster-Shafer theory of evidence is one of the major points of criticism
this formalism has to face. To overcome this difficulty various approximation
algorithms have been suggested that aim at reducing the number of focal
elements in the belief functions involved. This article reviews a number of
algorithms based on this method and introduces a new one—the DI algorithm—that
was designed to bring about minimal deviations in those values that are
relevant to decision making. It describes an empirical study that examines the
appropriateness of these approximation procedures in decision-making
situations. It presents and interprets the empirical findings along several
dimensions and discusses the various tradeoffs that have to be taken into
account when actually applying one of these methods.

debate

+ **The Dempster-Shafer theory
of evidence**

J Gordon, EH Shortliffe - B.G. Buchanan, E.H. Shortliffe (Eds.), Rule-Based Expert
Systems: The MYCIN Experiments of the Stanford Heuristic Programming Project,
Addison-Wesley, Reading, Mass (1984)

The drawbacks of pure probabilistic
methods and of the certainty factor model have led us in recent years to
consider alternate approaches. Particularly appealing is the mathematical
theory of evidence developed by Arthur Dempster. We are convinced it merits
careful study and interpretation in the context of expert systems. This theory
was first set forth by Dempster in the 1960s and subsequently extended by Glenn
Shafer. In 1976, the year after the first description of CF’s appeared, Shafer
published A Mathematical Theory of Evidence (Shafer, 1976). Its relevance to
the issues addressed in the CF model was not immediately recognized, but
recently researchers have begun to investigate applications of the theory to
expert systems (Barnett, 1981; Friedman, 1981; Garvey et al., 1981).

We believe that the advantage of the
Dempster-Shafer theory over previous approaches is its ability to model the
narrowing of the hypothesis set with the accumulation of evidence, a process
that characterizes diagnostic reasoning in medicine and expert reasoning in
general. An expert uses evidence that, instead of bearing on a single
hypothesis in the original hypothesis set, often bears on a larger subset of
this set. The functions and combining rule of the Dempster-Shafer theory are
well suited to represent this type of evidence and its aggregation.

multicriteria decision making

**The Dempster–Shafer theory
of evidence: an alternative approach to multicriteria decision modelling**

M Beynon, B Curry, P Morgan - Volume 28, Issue 1, February 2000, Pages 37–50

The objective of this paper is to describe the
potential offered by the Dempster–Shafer theory (DST) of evidence as a
promising improvement on “traditional” approaches to decision analysis.
Dempster–Shafer techniques originated in the work of Dempster on the use of
probabilities with upper and lower bounds. They have subsequently been
popularised in the literature on Artificial Intelligence (AI) and Expert
Systems, with particular emphasis placed on combining evidence from different
sources. In the paper we introduce the basic concepts of the DST of evidence,
briefly mentioning its origins and comparisons with the more traditional
Bayesian theory. Following this we discuss recent developments of this theory
including analytical and application areas of interest. Finally we discuss
developments via the use of an example incorporating DST with the Analytic
Hierarchy Process (AHP).

computation - approximation

+ **A computationally efficient approximation of Dempster-Shafer theory**

F Voorbraak - International Journal of Man-Machine Studies 30, May 1989, Pages
525–536

An often mentioned obstacle for the use of
Dempster-Shafer theory for the handling of uncertainty in expert systems is the
computational complexity of the theory. One cause of this complexity is the
fact that in Dempster-Shafer theory the evidence is represented by a belief
function which is induced by a basic probability assignment, i.e. a probability
measure on the powerset of possible answers to a question, and not by a
probability measure on the set of possible answers to a question, like in a
Bayesian approach. In this paper, we define a Bayesian approximation of a
belief function and show that combining the Bayesian approximations of belief
functions is computationally less involving than combining the belief functions
themselves, while in many practical applications replacing the belief functions
by their Bayesian approximations will not essentially affect the result.

debate - foundations

+ **Two views of belief: belief as generalized probability and belief as
evidence**

JY Halpern, R Fagin - Artificial intelligence, Volume 54, Issue 3, April 1992, Pages
275–317

Belief functions are mathematical objects defined
to satisfy three axioms that look somewhat similar to the Kolmogorov axioms
defining probability functions. We argue that there are (at least) two useful
and quite different ways of understanding belief functions. The first is as a
generalized probability function (which technically corresponds to the inner
measure induced by a probability function). The second is as a way of
representing *evidence*. Evidence, in turn, can be understood as a
mapping from probability functions to probability functions. It makes sense to
think of *updating* a belief if we think of it as a generalized
probability. On the other hand, it makes sense to *combine* two
beliefs (using, say, Dempster's *rule of combination*) only if we
think of the belief functions as representing evidence. Many previous papers
have pointed out problems with the belief function approach; the claim of this
paper is that these problems can be explained as a consequence of confounding
these two views of belief functions.

__Cited by 255____ __

debate - foundations

+ **Two views of belief: belief as generalized probability and belief as
evidence**

JY Halpern, R Fagin – Proc of AAAI, 112-119, 1990

Belief functions are mathematical objects defined
to satisfy three axioms that look somewhat similar to the Kolmogorov axioms
defining probability functions. We argue that there are (at least) two useful
and quite different ways of understanding belief functions. The first is as a
generalized probability function (which technically corresponds to the inner
measure induced by a probability function). The second is as a way of
representing *evidence*. Evidence, in turn, can be understood as a
mapping from probability functions to probability functions. It makes sense to
think of *updating* a belief if we think of it as a generalized
probability. On the other hand, it makes sense to *combine* two
beliefs (using, say, Dempster's *rule of combination*) only if we
think of the belief functions as representing evidence. Many previous papers
have pointed out problems with the belief function approach; the claim of this
paper is that these problems can be explained as a consequence of confounding
these two views of belief functions.

combination

T Denśux - Artificial Intelligence, Volume 172, Issues 2–3, February
2008, Pages 234–264

http://www.sciencedirect.com/science/article/pii/S0004370207001063

Dempster's rule plays a central role in the theory of belief
functions. However, it assumes the combined bodies of evidence to be distinct,
an assumption which is not always verified in practice. In this paper, a new
operator, the cautious rule of combination, is introduced. This operator is
commutative, associative and idempotent. This latter property makes it suitable
to combine belief functions induced by reliable, but possibly overlapping
bodies of evidence. A dual operator, the bold disjunctive rule, is also
introduced. This operator is also commutative, associative and idempotent, and
can be used to combine belief functions issues from possibly overlapping *and* unreliable
sources. Finally, the cautious and bold rules are shown to be particular
members of infinite families of conjunctive and disjunctive combination rules
based on triangular norms and conorms.

advances – expert systems - hierarchical

+ **A method for managing evidential reasoning in a hierarchical hypothesis
space**

J Gordon, EH Shortliffe - Artificial Intelligence, Volume 26, Issue 3, July 1985, Pages
323–357

http://www.sciencedirect.com/science/article/pii/0004370285900645

Although informal models of evidential reasoning
have been successfully applied in automated reasoning systems, it is generally
difficult to define the range of their applicability. In addition, they have
not provided a basis for consistent management of evidence bearing on
hypotheses that are related hierarchically. The Dempster-Shafer (D-S) theory of
evidence is appealing because it does suggest a coherent approach for dealing
with such relationships. However, the theory's complexity and potential for
computational inefficiency have tended to discourage its use in reasoning
systems. In this paper we describe the central elements of the D-S theory,
basing our exposition on simple examples drawn from the field of medicine. We
then demonstrate the relevance of the D-S theory to a familiar expert-system
domain, namely the bacterial-organism identification problem that lies at the
heart of the mycin system. Finally, we present a new adaptation of
the D-S approach that achieves computational efficiency while permitting the
management of evidential reasoning within an abstraction hierarchy.

Conflict - combination

+ **Analyzing the degree of conflict among belief functions**

W Liu - Artificial Intelligence, Volume 170, Issue 11, August 2006,
Pages 909–924

http://www.sciencedirect.com/science/article/pii/S0004370206000658

The study of alternative combination rules in DS
theory when evidence is in conflict has emerged again recently as an
interesting topic, especially in data/information fusion applications. These
studies have mainly focused on investigating which alternative would be appropriate
for which conflicting situation, under the assumption that a *conflict* is
identified. The issue of detection (or identification) of conflict among
evidence has been ignored. In this paper, we formally define when two basic
belief assignments are in conflict. This definition deploys quantitative
measures of both the mass of the combined belief assigned to the emptyset
before normalization and the distance between betting commitments of beliefs.
We argue that only when both measures are high, it is safe to say the evidence
is in conflict. This definition can be served as a prerequisite for selecting
appropriate combination rules.

debate

+ **Perspectives on the theory and practice of belief functions**

G Shafer - International Journal of Approximate Reasoning, Volume 4,
Issues 5–6, September–November 1990, 323–362

The theory of belief functions is a generalization
of the Bayesian theory of subjective probability judgement. The author's 1976
book, A Mathematical Theory of Evidence, is still a standard reference for this
theory, but it is concerned primarily with mathematical foundations. Since
1976, considerable work has been done on interpretation and implementation of
the theory. This article reviews this work, as well as newer work on
mathematical foundations. It also considers the place of belief functions within
the broader topic of probability and the place of probability within the larger
set of formalisms used by artificial intelligence.

Advances - hierarchical

+ **Implementing Dempster's rule for hierarchical evidence**

G Shafer, R Logan - Artificial Intelligence, Volume 33, Issue 3,
November 1987, Pages 271–298

This article gives an algorithm for the exact
implementation of Dempster's rule in the case of hierarchical evidence. This
algorithm is computationally efficient, and it makes the approximation
suggested by Gordon and Shortliffe unnecessary. The algorithm itself is simple,
but its derivation depends on a detailed understanding of the interaction of
hierarchical evidence.

combination

+ **On the justification of Dempster's rule of combination**

F Voorbraak - Artificial Intelligence 48, Issue 2, March 1991, Pages 171–197

http://www.sciencedirect.com/science/article/pii/000437029190060W

In Dempster-Shafer theory it is claimed that the
pooling of evidence is reflected by Dempster's rule of combination, provided
certain requirements are met. The justification of this claim is problematic,
since the existing formulations of the requirements for the use of Dempster's
rule are not completely clear. In this paper, randomly coded messages, Shafer's
canonical examples for Dempster-Shafer theory, are employed to clarify these
requirements and to evaluate Dempster's rule. The range of applicability of
Dempster-Shafer theory will turn out to be rather limited. Further, it will be
argued that the mentioned requirements do not guarantee the validity of the
rule and some possible additional conditions will be described.

advances - information

- **Properties of measures of information in evidence and possibility
theories**

D Dubois, H Prade - Fuzzy Sets and Systems 24, Issue 2, November 1987, Pages 161–182

http://www.sciencedirect.com/science/article/pii/S0165011499800050

An overview of information measures recently
introduced by several authors in the setting of Shafer's theory of evidence is
proposed. New results pertaining to additivity and monotonicity properties of
these measures of information are presented. The interpretation of each measure
of information as opposed to others is discussed. The potential usefulness of
measures of specificity or imprecision is suggested, and a ‘principle of
minimal specificity' is stated for the purpose of reconstructing a body of
evidence from incomplete knowledge.

applications – document retrieval

+ **Dempster-Shafer's theory of evidence applied to
structured documents: modelling uncertainty**

M Lalmas - Proceedings of the 20th annual international ACM SIGIR
conference on Research and development in information retrieval, pages 110-118,
1997

Documents often display a structure determined by
the author, e.g., several chapters, each with several sub-chapters and so on.
Taking into account the structure of a document allows the retrieval process to
focus on those parts of the documents that are most relevant to an information
need. Chiaramella et al advanced a model for indexing and retrieving structured
documents. Their aim was to express the model within a framework based on
formal logics with associated theories. They developed the logical formalism of
the model. This paper adds to this model a theory of uncertainty, the
Dempster-Shafer theory of evidence. It is shown that the theory provides a
rule, the Dempster’s combination rule, that allows the expression of the
uncertainty with respect to parts of a document, and that is compatible with
the logical model developed by Chiaramella et al.

other theories – rough sets

+ **Knowledge reduction in random information systems via
Dempster–Shafer theory of evidence**

WZ Wu, M Zhang, HZ Li, JS Mi - Information Sciences 174, Issues 3–4, 11
August 2005, Pages 143–164

http://www.sciencedirect.com/science/article/pii/S0020025504002610

Knowledge reduction is one of the main problems in
the study of rough set theory. This paper deals with knowledge reduction in
(random) information systems based on Dempster–Shafer theory of evidence. The
concepts of belief and plausibility reducts in (random) information systems are
first introduced. It is proved that both of belief reduct and plausibility reduct
are equivalent to classical reduct in (random) information systems. The
relative belief and plausibility reducts in consistent and inconsistent
(random) decision systems are then proposed and compared to the relative reduct
and relationships between the new reducts and some existing ones are examined.

decision making
– frameworks – probability transformation - TBM

+ **Decision making in
the TBM: the necessity of the pignistic transformation**

P Smets -
International Journal of Approximate Reasoning, Volume 38, Issue 2, February
2005, Pages 133–147

http://www.sciencedirect.com/science/article/pii/S0888613X04000593

In the transferable belief
model (TBM), pignistic probabilities are used for decision making. The nature
of the pignistic transformation is justified by a linearity requirement. We
justify the origin of this requirement showing it is not ad hoc but unavoidable
provides one accepts expected utility theory.

other theories – rough sets

+ **Interpretations of
belief functions in the theory of rough sets**

YY Yao, PJ Lingras - Information Sciences 104, Issues 1–2,
January 1998, Pages 81–106

This paper reviews and examines
interpretations of belief functions in the theory of rough sets with finite
universe. The concept of standard rough set algebras is generalized in two
directions. One is based on the use of nonequivalence relations. The other is
based on relations over two universes, which leads to the notion of interval
algebras. Pawlak rough set algebras may be used to interpret belief functions
whose focal elements form a partition of the universe. Generalized rough set
algebras using nonequivalence relations may be used to interpret belief
functions which have less than |*U*| focal elements, where |*U*| is
the cardinality of the universe *U* on which belief functions
are defined. Interval algebras may be used to interpret any belief functions.

machine learning - classification

E Mandler and J Schurmann – In
Gelsema, E., & Kanal, L. (Eds.), Pattern Recognition and Artificial
Intelligence, 381-393, 1988

no
abstract

combination

**+ Combination
of evidence in Dempster-Shafer theory**

K Sentz, S Ferson – Technical Report SAND 2002-0835, SANDIA, April 2002

Dempster-Shafer
theory offers an alternative to traditional probabilistic theory for the
mathematical representation of uncertainty. The significant innovation of this
framework is that it allows for the allocation of a probability mass to sets or
intervals. Dempster-Shafer theory does not require an assumption regarding the
probability of the individual constituents of the set or interval. This is a
potentially valuable tool for the evaluation of risk and reliability in
engineering applications when it is not possible to obtain a precise
measurement from experiments, or when knowledge is obtained from expert
elicitation. An important aspect of this theory is the combination of evidence
obtained from multiple sources and the modeling of conflict between them. This
report surveys a number of possible combination rules for Dempster-Shafer
structures and provides examples of the implementation of these rules for
discrete and interval-valued data.

Cited by __742__

advances – information measure

- **Uniqueness of information measure in the theory of evidence**

A Ramer - Fuzzy Sets and Systems, Volume 24, Issue 2, November 1987,
Pages 183–196

http://www.sciencedirect.com/science/article/pii/0165011487900893

An evidence distribution on a
set *X* assigns non-negative weights to the subsets of *X*.
Such weights must sum to one and the empty set is given weight 0. An
information measure can be defined for such an evidence distribution.

If *mi* are the
weights assigned to subsets *Ai*, and *ai* are the
cardinalities of these subsets, then the function.*Σmi* log *ai* satisfies
all the usual axioms of an information measure. In this paper we show that,
conversely, these axioms are sufficient to characterize uniquely the above
measure. It can be thus considered as the main uncertainty function for the
theory of evidence.

We demonstrate that using only the
properties of symmetry, additivity and subadditivity the problem of uniqueness
can be reduced to finding linear functionals on the space of functions analytic
at origin. We surmise that under a suitable continuity hypothesis, all such
functionals can be represented as linear combinations of the coefficients of
Taylor series. Our function then represents the first derivative evaluated at
0. An alternative approach is then discussed. We assume a form of branching
property, suggested by the monotonicity considerations. Now the properties of
symmetry, additivity and subadditivity, together with branching again offer the
unique characterization of the information function. No continuity assumption
whatsoever is needed and the proof is entirely elementary.

other theories – possibility - logic

**+ Possibilistic
logic**

D Dubois, J Lang, H Prade - 1994

Possibilistic logic is a logic of uncertainty
tailored for reasoning under incomplete evidence and partially inconsistent
knowledge. At the syntactic level it handles formulas of propositional or
first-order logic to which are attached numbers between 0 and 1, or more
generally elements in a totally ordered set. These weights are lower bounds on
so-called degrees of necessity or degrees of possibility of the corresponding
formulas. The degree of necessity (or certainty) of a formula expresses to what
extent the available evidence entails the truth of this formula. The degree of
possibility expresses to what extent the truth of the formula is not
incompatible with the available evidence.

At the mathematical level, degrees of possibility
and necessity are closely related to fuzzy sets (Zadeh, 1965, 1978a), and
possibilistic logic is especially adapted to automated reasoning when the
available information is pervaded with vagueness. A vague piece of evidence can
be viewed as defining an implicit ordering on the possible worlds it refers to,
this ordering being encoded by means of fuzzy set membership functions. Hence
possibilistic logic is a tool for reasoning under uncertainty based on the idea
of (complete) ordering rather than counting, contrary to probabilistic logic.
To figure out how possibilistic logic could emerge as a worth-studying
formalism, it might be interesting to go back to the origins of fuzzy set
theory and what is called "fuzzy logic". Fuzzy sets were introduced
by Zadeh (1965) in an attempt to propose a mathematical tool describing the
type of model people use when reasoning about systems.

graphical models – computation – propagation - combination

+ **Propagating belief functions in qualitative Markov trees**

G Shafer, PP Shenoy, K Mellouli - International Journal of Approximate Reasoning, Volume 1,
Issue 4, 1987, 349–400

http://www.sciencedirect.com/science/article/pii/0888613X87900247

This article
is concerned with the computational aspects of combining evidence within the
theory of belief functions. It shows that by taking advantage of logical or
categorical relations among the questions we consider, we can sometimes avoid
the computational complexity associated with brute-force application of
Dempster's rule.

The mathematical
setting for this article is the lattice of partitions of a fixed overall frame
of discernment. Different questions are represented by different partitions of
this frame, and the categorical relations among these questions are represented
by relations of qualitative conditional independence or dependence among the
partitions. Qualitative conditional independence is a categorical rather than a
probabilistic concept, but it is analogous to conditional independence for
random variables. We show that efficient implementation of Dempster's rule is
possible if the questions or partitions for which we have evidence are arranged
in a qualitative Markov tree—a tree in which separations indicate relations of
qualitative conditional independence. In this case, Dempster's rule can be
implemented by propagating belief functions through the tree.

Applications – biomedical engineering

+ **Sleep staging automaton based on the theory of evidence**

JC Principe, SK Gala, TG Chang - IEEE Transactions on Biomedical Engineering,
Volume 36, Issue 5, 503 – 509, 1989

The authors address sleep staging as a medical
decision problem. They develop a model for automated sleep staging by combining
signal information, human heuristic knowledge in the form of rules, and a
mathematical framework. The EEG/EOG/EMG
(electroencephalogram/electroculogram/electromyogram) events relevant for sleep
staging are detected in real time by an existing front-end system and are
summarized per minute. These token data are translated, normalized and
constitute the input alphabet to a finite-state machine (automaton). The
processed token events are used as partial belief in a set of anthropomimetic
rules, which encode human knowledge about the occurrence of a particular sleep
stage. The Dempster-Shafer theory of evidence weighs the partial beliefs and
attributes the minute sleep stage to the machine state transition that displays
the highest final belief. Results are briefly presented.

Cited by __83__

machine learning - classification

+ **A new technique for combining
multiple classifiers using the Dempster-Shafer theory of evidence**

A Al-Ani, M Deriche - Journal of Artificial Intelligence Research, Volume 17, pages
333-361, 2002

This paper presents a new classifier combination
technique based on the Dempster-Shafer theory of evidence. The Dempster-Shafer
theory of evidence is a powerful method for combining measures of evidence from
different classifiers. However, since each of the available methods that
estimates the evidence of classifiers has its own limitations, we propose here
a new implementation which adapts to training data so that the overall mean
square error is minimized. The proposed technique is shown to outperform most
available classifier combination methods when tested on three different
classification problems.

graphical models - markov

+ **Multisensor triplet Markov chains and theory of evidence**

W Pieczynski - International Journal of Approximate Reasoning, Volume
45, Issue 1, May 2007, Pages 1–16

Hidden Markov chains (HMC) are widely
applied in various problems occurring in different areas like Biosciences,
Climatology, Communications, Ecology, Econometrics and Finances, Image or
Signal processing. In such models, the hidden process of interest *X* is
a Markov chain, which must be estimated from an observable *Y*,
interpretable as being a noisy version of *X*. The success of HMC is
mainly due to the fact that the conditional probability distribution of the
hidden process with respect to the observed process remains Markov, which makes
possible different processing strategies such as Bayesian restoration. HMC have
been recently generalized to “Pairwise” Markov chains (PMC) and “Triplet”
Markov chains (TMC), which offer similar processing advantages and superior
modeling capabilities. In PMC, one directly assumes the Markovianity of the
pair (*X*, *Y*) and in TMC, the distribution of the pair (*X*, *Y*)
is the marginal distribution of a Markov process (*X*, *U*, *Y*),
where *U* is an auxiliary process, possibly contrived.
Otherwise, the Dempster–Shafer fusion can offer interesting extensions of the
calculation of the “a posteriori” distribution of the hidden data.

The aim of this paper is to present
different possibilities of using the Dempster–Shafer fusion in the context of
different multisensor Markov models. We show that the posterior distribution
remains calculable in different general situations and present some examples of
their applications in remote sensing area.

Cited by __61__

combination

+ **Dempster's rule of combination is #P-complete**

P Orponen - Artificial Intelligence, Volume 44, Issues 1–2, July 1990,
Pages 245–253

http://www.sciencedirect.com/science/article/pii/0004370290901037

We consider the complexity of combining bodies of
evidence according to the rules of the Dempster-Shafer theory of evidence. We
prove that, given as input a set of tables representing basic probability
assignments m_{1}, …, m_{n} over a frame of discernment Θ,
and a set *A* ⊆ *Θ*,
the problem of computing the combined basic probability value (m_{1} ⊕ … ⊕ m_{n})(A) *is* #*P*-complete.
As a corollary, we obtain that while the simple belief, plausibility, and
commonality values *Bel*(A), *Pl*(A), and Q(A) can be
computed in polynomial time, the problems of computing the combinations (Bel_{1} ⊕ … ⊕ Bel_{n}(A), (Pl_{1} ⊕ … ⊕ Pl_{n})(A), and (Q_{1} ⊕ … ⊕ Q_{n})(A) are #*P*-complete.

consonant approximation

+ **Consonant approximations of belief functions**

D Dubois, H Prade - International Journal of Approximate Reasoning, Volume 4, Issues
5–6, September–November 1990, Pages 419–449

A general notion of approximation of a belief
function by some other set function is introduced that is based on a recently
introduced definition of inclusion between random sets. Viewing a fuzzy set as
a consonant random set, it is shown how to construct fuzzy sets that may act as
approximations of belief functions. Two kinds of approximations are considered:
inner approximations that provide upper bounds on belief degrees and lower
bounds on plausibility degrees, and outer approximations that provide lower
bounds on belief degrees and upper bounds on plausibility degrees. Minimal
outer and maximal inner consonant approximations are characterized in a
constructive way. The particular problem of approximating a probability measure
by a fuzzy set is solved. Applications to the approximate computation of belief
functions on Cartesian products, combinations by Dempster's rule, and functions
of random-set-valued arguments by means of fuzzy set operations are sketched.

geometry – frameworks - combination

+ **A geometric approach to the theory
of evidence**

F Cuzzolin - Systems, Man, and Cybernetics, Part C, Vol. 38, Issue 4, pages
522-534, 2008

In this paper, we propose a geometric approach to
the theory of evidence based on convex geometric interpretations of its two key
notions of belief function (b.f.) and Dempster's sum. On one side, we analyze
the geometry of b.f.'s as points of a polytope in the Cartesian space called
belief space, and discuss the intimate relationship between basic probability
assignment and convex combination. On the other side, we study the global
geometry of Dempster's rule by describing its action on those convex
combinations. By proving that Dempster's sum and convex closure commute, we are
able to depict the geometric structure of conditional subspaces, i.e., sets of
b.f.'s conditioned by a given function b. Natural applications of these
geometric methods to classical problems such as probabilistic approximation and
canonical decomposition are outlined.

Cited by __94__

debate

+ **Reasoning with imprecise belief structures**

T Denśux - International Journal of Approximate Reasoning, Volume 20,
Issue 1, January 1999, Pages 79–111

http://www.sciencedirect.com/science/article/pii/S0888613X00889446

This
paper extends the theory of belief functions by introducing new concepts and
techniques, allowing to model the situation in which the beliefs held by a
rational agent may only be expressed (or are only known) with some imprecision.
Central to our approach is the concept of interval-valued belief structure
(IBS), defined as a set of belief

structures
verifying certain constraints. Starting from this definition, many other
concepts of Evidence Theory (including belief and plausibility functions,
pignistic probabilities, combination rules and uncertainty measures) are generalized
to cope with imprecision in the belief numbers attached to each hypothesis. An
application of this new framework to the classification of patterns with
partially known feature values is demonstrated.

combination

- **On the evidence inference theory**

YG Wu, JY Yang, LJ Liu - Information Sciences, Volume 89, Issues 3–4,
March 1996, Pages 245–260

http://www.sciencedirect.com/science/article/pii/002002559500226X

The Dempster-Shafer theory of evidence reasoning
(D-S theory) has been widely discussed and used recently, because it is a
reasonable, convenient, and promising method to combine uncertain information
from disparate sources with different levels of abstraction. On the other hand,
the D-S theory has sparked considerable debate among statisticians and
knowledge engineers. The theory has been criticized and debated upon its
behavior and attributes, such as high computational complexity, evidence
independency requirement in its combination rule, etc. some principal problems
of the D-S theory are discussed in the paper. The relationship of the D-S
theory and the classical probability theory is analyzed first, and then a
generalized evidence combination formula relaxing the requirement of evidence
independency is presented, which makes the D-S theory more realistic to
applications.

other theories – rough sets

**Connections between rough set theory and Dempster-Shafer theory of evidence**

WZ Wu, YEE Leung, WX Zhang - International Journal of General Systems, Volume
31, Issue 4, pages 405-430, 2002

http://www.tandfonline.com/doi/abs/10.1080/0308107021000013626

In rough set theory there exists a pair of
approximation operators, the upper and lower approximations, whereas in
Dempster-Shafer theory of evidence there exists a dual pair of uncertainty
measures, the plausibility and belief functions. It seems that there is some
kind of natural connection between the two theories. The purpose of this paper
is to establish the relationship between rough set theory and Dempster-Shafer
theory of evidence. Various generalizations of the Dempster-Shafer belief
structure and their induced uncertainty measures, the plausibility and belief
functions, are first reviewed and examined. Generalizations of Pawlak
approximation space and their induced approximation operators, the upper and
lower approximations, are then summarized. Concepts of random rough sets, which
include the mechanisms of numeric and non-numeric aspects of uncertain
knowledge, are then proposed. Notions of the Dempster-Shafer theory of evidence
within the framework of rough set theory are subsequently formed and
interpreted. It is demonstrated that various belief structures are associated
with various rough approximation spaces such that different dual pairs of upper
and lower approximation operators induced by the rough approximation spaces may
be used to interpret the corresponding dual pairs of plausibility and belief
functions induced by the belief structures.

geometry - distance

+ **A new distance between two bodies of evidence**

AL Jousselme, D Grenier, É Bossé - Information fusion, Volume 2, Issue
2, June 2001, Pages 91–101

http://www.sciencedirect.com/science/article/pii/S1566253501000264

We present a measure of performance (MOP) for
identification algorithms based on the evidential theory of Dempster–Shafer. As
an MOP, we introduce a principled distance between two basic probability
assignments (BPAs) (or two bodies of evidence) based on a quantification of the
similarity between sets. We give a geometrical interpretation of BPA and show
that the proposed distance satisfies all the requirements for a metric. We also
show the link with the quantification of Dempster's weight of conflict proposed
by George and Pal. We compare this MOP to that described by Fixsen and Mahler
and illustrate the behaviors of the two MOPs with numerical examples.

other theories - intervals

+ **Uncertain inference using interval probability theory**

JW Hall, DI Blockley, JP Davis - International Journal of Approximate Reasoning,
Volume 19, Issues 3–4, October–November 1998, Pages 247–264

http://www.sciencedirect.com/science/article/pii/S0888613X98100105

The use of interval probability theory (IPT) for
uncertain inference is demonstrated. The general inference rule adopted is the
theorem of total probability. This enables information on the relevance of the
elements of the power set of evidence to be combined with the measures of the
support for and dependence between each item of evidence. The approach
recognises the importance of the structure of inference problems and yet is an
open world theory in which the domain need not be completely specified in order
to obtain meaningful inferences. IPT is used to manipulate conflicting evidence
and to merge evidence on the dependability of a process with the data handled
by that process. Uncertain inference using IPT is compared with Bayesian
inference.

__Cited by 66____ __

decision making - linguistic

JM Merigó, M Casanovas, L Martínez - International Journal of Uncertainty, Fuzziness and Knowledge-Based
Systems, Volume 18, Issue 03, page 287, June 2010

http://www.worldscientific.com/doi/abs/10.1142/S0218488510006544?src=recsys

In this paper, we develop a new approach for
decision making with Dempster-Shafer theory of evidence by using linguistic
information. We suggest the use of different types of linguistic aggregation
operators in the model. We then obtain as a result, the belief structure —
linguistic ordered weighted averaging (BS-LOWA), the BS — linguistic hybrid
averaging (BS-LHA) and a wide range of particular cases. Some of their main
properties are studied. Finally, we provide an illustrative example that shows
the different results obtained by using different types of linguistic
aggregation operators in the new approach.

dissemination

G Shafer - Encyclopedia of Artificial
Intelligence, 1992

The Dempster-Shafer theory, also known as the
theory of belief functions, is a generalization of the Bayesian theory of
subjective probability. Whereas the Bayesian theory requires probabilities for
each question of interest, belief functions allow us to base degrees of belief
for one question on probabilities for a related question. These degrees of
belief may or may not have the mathematical properties of probabilities; how
much they differ from probabilities will depend on how closely the two
questions are related.

The Dempster-Shafer theory owes its name to work by
A. P. Dempster (1968) and Glenn Shafer (1976), but the kind of reasoning the
theory uses can be found as far back as the seventeenth century. The theory
came to the attention of AI researchers in the early 1980s, when they were
trying to adapt probability theory to expert systems. Dempster-Shafer degrees
of belief resemble the certainty factors in MYCIN, and this resemblance
suggested that they might combine the rigor of probability theory with the flexibility
of rule-based systems. Subsequent work has made clear that the management of
uncertainty inherently requires more structure than is available in simple
rule-based systems, but the Dempster-Shafer theory remains attractive because
of its relative flexibility

Cited by __72 __

machine learning – neural networks - classification

+ **Combining the results of several neural network classifiers**

G Rogova - Neural networks, Volume 7, Issue 5, 1994, Pages 777–781

http://www.sciencedirect.com/science/article/pii/089360809490099X

Neural networks and traditional classifiers work
well for optical character recognition; however, it is advantageous to combine
the results of several algorithms to improve classification accuracies. This
paper presents a combination method based on the Dempster-Shafer theory of
evidence, which uses statistical information about the relative classification
strengths of several classifiers. Numerous experiments show the effectiveness
of this approach. Our method allows 15–30% reduction of misclassification error
compared to the best individual classifier.

Cited by __408____ __

graphical models - propagation

__On the
propagation of beliefs in networks using the Dempster-Shafer theory of evidence__

K Mellouli - 1987 - University of Kansas, Business

http://libra.msra.cn/Publication/1370738/on-the-propagation-of-beliefs-in-networks-using-the-dempster-shafer-theory-of-evidence

no
abstract

statistics – expert systems

+ **A statistical viewpoint on the theory of evidence**

RA Hummel, MS Landy - Pattern Analysis and Machine Intelligence, Vol 10, No 2, pages
235-247, 1988

The authors provide a perspective and
interpretation regarding the Dempster-Shafer theory of evidence that regards
the combination formulas as statistics of the opinions of experts. This is done
by introducing spaces with binary operations that are simpler to interpret or
simpler to implement than the standard combination formula, and showing that
these spaces can be mapped homomorphically onto the Dempster-Shafer
theory-of-evidence space. The experts in the space of opinions-of-experts
combine information in a Bayesian fashion. Alternative spaces for the
combination of evidence suggested by this viewpoint are presented.

Machine learning – classification – applications - medical

I Bloch - Pattern Recognition Letters, Volume 17, Issue 8, 1 July 1996,
Pages 905–919

This paper points out some key features of
Dempster-Shafer evidence theory for data fusion in medical imaging. Examples
are provided to show its ability to take into account a large variety of
situations, which actually often occur and are not always well managed by
classical approaches nor by previous applications of Dempster-Shafer theory in
medical imaging. The modelization of both uncertainty and imprecision, the
introduction of possible partial or global ignorance, the computation of
conflict between images, the possible introduction of a priori information are
all powerful aspects of this theory, which deserve to be more exploited in
medical image processing. They may be of great influence on the final decision.
They are illustrated on a simple example for classifying brain tissues in pathological
dual echo MR images. In particular, partial volume effect can be properly
managed by this approach.

advances – combination – generalized Bayesian theorem

+ **Belief functions: the disjunctive rule of combination and the
generalized Bayesian theorem**

P Smets - International Journal of Approximate Reasoning 9, Issue 1,
August 1993, Pages 1–35

http://www.sciencedirect.com/science/article/pii/0888613X9390005X

We generalize the Bayes' theorem within the
transferable belief model framework. The generalized Bayesian theorem (GBT)
allows us to compute the belief over a space θ given an observation x ⊆ X when one knows only the
beliefs over X for every θi
∈ Θ. We also discuss the
disjunctive rule of combination (DRC) for distinct pieces of evidence. This
rule allows us to compute the belief over X from the beliefs induced by two
distinct pieces of evidence when one knows only that one of the pieces of
evidence holds. The properties of the DRC and GBT and their uses for belief
propagation in directed belief networks are analyzed. The use of the
discounting factors is justified. The application of these rules is illustrated
by an example of medical diagnosis.

combination

+ **A new combination of evidence based on compromise**

K Yamada - Fuzzy sets and Systems 159, Issue 13, 1 July 2008, Pages
1689–1708

http://www.sciencedirect.com/science/article/pii/S0165011407005222

yamada@kjs.nagaokaut.ac.jp

Dempster rule of combination is the standard way of
combining multiple pieces of evidence given by independent sources of
information. However, it aroused many controversies about its validity, and
many alternatives have been proposed. The paper examines the model of
combination in Dempster's original paper and indicates that handling of the
independence required among multiple pieces of evidence is strange from the
viewpoint of semantics, where the independence among occurrences of multiple
pieces of information might be confused with the consistency among contents of
the information. The paper then proposes a new model of combination and a new
rule of combination called combination by compromise as a consensus generator.
The properties of the proposed combination as well as several alternative
combination methods proposed so far are discussed in the light of the drawbacks
and advantages of Dempster rule. Several numerical examples which demonstrate
the properties are also shown. The discussion and the examples suggest that the
proposed combination produces the most preferable results among them from the
viewpoints of consensus generation.

information measure - entropy

**Measures of entropy in the theory
of evidence**

MT Lamata and S Moral - International Journal of General System 14, Issue 4, pages
297-305, 1988

http://www.tandfonline.com/doi/abs/10.1080/03081078808935019

This paper considers two measures of entropy for
the Theory of Evidence and studies their properties. The first measure is based
on the measures of entropy and specificity as defined by Yager. The other one
may easily be applied to more general classes of fuzzy measures.

combination – TBM - frameworks

+ **The combination of evidence in the transferable belief model**

P Smets - Pattern Analysis and Machine Intelligence, Vol 12, No 5, pages
447-458, 1990

A description of the transferable belief model, which is used to
quantify degrees of belief based on belief functions, is given. The impact of
open- and closed-world assumption on conditioning is discussed. The nature of
the frame of discernment on which a degree of belief will be established is
discussed. A set of axioms justifying Dempster's rule for the combination of
belief functions induced by two distinct evidences is presented.

Cited by __924__

applications – acoustics - classification

+ **Combination of Acoustic Classifiers Based on Dempster-Shafer Theory of Evidence**

F Valente, H Hermansky - ICASSP 2007

In this paper we investigate combination of neural
net based classifiers using Dempster-Shafer Theory of Evidence. Under some
assumptions, combination rule resembles a product of errors rule observed in
human speech perception. Different combination are tested in ASR experiments
both in matched and mismatched conditions and compared with more conventional
probability combination rules. Proposed techniques are particularly effective
in mismatched conditions.

other theories – model logic

- **On modal logic interpretation of Dempster–Shafer theory of evidence**

D Harmanec, GJ Klir, G Resconi - International Journal of Intelligent
Systems, Vol 9, No 10, pages 941-951, 1994

http://onlinelibrary.wiley.com/doi/10.1002/int.4550091003/abstract

This article further develops one branch of
research initiated in an article by Resconi, Klir, and St. Clair (G. Resconi,
G. J. Klir, and U. St. Clair, *Int. J. Gen. Syst*., **21**(1),
23-50 (1992) and continued in another article by Resconi et al. (*Int. J.
Uncertainty, Fuzziness and Knowledge-Based Systems*, **1**(1),
1993). It fully formulates an interpretation of the Dempster-Shafer theory in
terms of the standard semantics of modal logic. It is shown how to represent
the basic probability assignment function as well as the commonality function
of the Dempster-Shafer theory by modal logic and that this representation is
complete for rational-valued functions (basic assignment, belief, or
plausibility functions).

other theories – rough sets

+ **Rough mereology: A new paradigm for approximate reasoning**

L Polkowski, A Skowron - International Journal of Approximate Reasoning, Volume 15, Issue
4, November 1996, Pages 333–365

http://www.sciencedirect.com/science/article/pii/S0888613X96000722

We are concerned with formal models of reasoning
under uncertainty. Many approaches to this problem are known in the literature:
Dempster-Shafer theory, bayesian-based reasoning, belief networks, fuzzy
logics, etc. We propose rough mereology as a foundation for approximate
reasoning about complex objects. Our notion of a complex object includes
approximate proofs understood as schemes constructed to support our assertions
about the world on the basis of our incomplete or uncertain knowledge.

applications - measurement

**[book]
****Measurement uncertainty: An approach via the mathematical theory of evidence**

S Salicone - Springer Science & Business Media, 4 Jun 2007

simona.salicone@polimi.it

The expression of uncertainty in measurement is a
challenging aspect for researchers and engineers working in instrumentation and
measurement because it involves physical, mathematical and philosophical
issues. This problem is intensified by the limitations of the probabilistic
approach used by the current standard (GUM). This text is the first to make
full use of the mathematical theory of evidence to express the uncertainty in
measurements. It gives an overview of the current standard, then pinpoints and
constructively resolves its limitations through its unique approach. The text
presents various tools for evaluating uncertainty, beginning with the
probabilistic approach and concluding with the expression of uncertainty using
random-fuzzy variables. The exposition is driven by numerous examples. The book
is designed for immediate use and application in research and laboratory work.
Apart from a classroom setting, this book can be used by practitioners in a
variety of fields (including applied mathematics, applied probability,
electrical and computer engineering, and experimental physics), and by such
institutions as the IEEE, ISA, and National Institute of Standards and
Technology.

advances - frames

+ **Theory of evidence ****and non-exhaustive frames of discernment: Plausibilities correction
methods**

F Janez, A Appriou - International Journal of Approximate Reasoning 18, No 1–2,
January–February 1998, Pages 1–19

http://www.sciencedirect.com/science/article/pii/S0888613X97100019

Benefits gained by a multisource solution in
various contexts of application are nowadays obvious. The goal of this
interesting approach is both to capture benefits of sources and to minimize
their limitations. Usually, each source is defined and modeled over a unique
frame composed of the hypotheses to discern. Sources can then be merged by the
combination process provided by the theory that enabled their modeling. On the
other hand, this process is no more applicable to sources defined on different
frames in terms of the hypotheses they consider. It is the case for example of
two sources defined respectively on the frames {H1, H2} and {H2m H3}. This
problem although frequently encountered in the development of operational systems
has paradoxically not been extensively treated. In a previous article, we have
already presented methods mainly based on a technique called “deconditioning”
and that allow the combination of such sources. They are developed in the
theory of evidence's framework, a priori the most appropriate for this problem.
We complete our investigation by proposing in this article other methods based
on the same framework.

debate - statistics

+ **Confidence factors, empiricism
and the Dempster-Shafer theory of
evidence**

JF Lemmer - arXiv preprint arXiv:1304.3437, 2013

The issue of confidence factors in Knowledge Based
Systems has become increasingly important and Dempster-Shafer (DS) theory has
become increasingly popular as a basis for these factors. This paper discusses
the need for an empirical lnterpretatlon of any theory of confidence factors
applied to Knowledge Based Systems and describes an empirical lnterpretatlon of
DS theory suggesting that the theory has been extensively misinterpreted. For
the essentially syntactic DS theory, a model is developed based on sample
spaces, the traditional semantic model of probability theory. This model is
used to show that, if belief functions are based on reasonably accurate
sampling or observation of a sample space, then the beliefs and upper
probabilities as computed according to DS theory cannot be interpreted as frequency
ratios. Since many proposed applications of DS theory use belief functions in
situations with statistically derived evidence (Wesley [1]) and seem to appeal
to statistical intuition to provide an lnterpretatlon of the results as has
Garvey [2], it may be argued that DS theory has often been misapplied.

combination - geometry

+ **Combining belief functions based on distance of evidence**

D Yong, S WenKang, Z ZhenFu, L Qi - Decision support systems, Volume 38, Issue
3, December 2004, Pages 489–493

http://www.sciencedirect.com/science/article/pii/S0167923604001447

A modified average method to combine belief
function based on distance measures of evidence is proposed. The weight of each
body of evidence (BOE) is taken into account. A numerical example is shown to
illustrate the use of the proposed method to combine conflicting evidence. Some
open issues are discussed in the final section.

applications – information retrieval

**[thesis]
Theories**

M Lalmas - University of Glasgow, 1996

http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.307178

other theories – fuzzy - advances

+ **Modeling vague beliefs using fuzzy-valued belief structures**

T Denśux - Fuzzy Sets and Systems, Volume 116, Issue 2, 1 December 2000,
Pages 167–199

http://www.sciencedirect.com/science/article/pii/S0165011498004059

This paper presents a rational approach to the
representation and manipulation of imprecise degrees of belief in the framework
of evidence theory. We adopt as a starting point the non-probabilistic
interpretation of belief functions provided by Smets’ Transferable Belief
Model, as well as previous generalizations of evidence theory allowing to deal
with fuzzy propositions. We then introduce the concepts of interval-valued and
fuzzy-valued belief structures, defined, respectively, as crisp and fuzzy sets
of belief structures verifying hard or elastic constraints. We then proceed with
a generalization of various concepts of Dempster–Shafer theory including those
of belief and plausibility functions, combination rules and normalization
procedures. Most calculations implied by the manipulation of these concepts are
based on simple forms of linear programming problems for which analytical
solutions exist, making the whole scheme computationally tractable. We discuss
the application of this framework in the areas of decision making under
uncertainty and classification of fuzzy data.

combination – expert systems

- **Evidence combination in expert systems**

L Lesmo, L Saitta, P Torasso - International Journal of Man-Machine
Studies 22, Issue 3, March 1985, Pages 307–326

http://www.sciencedirect.com/science/article/pii/S0020737385800067

This paper
discusses some of the problems related to the representation of uncertain
knowledge and to the combination of evidence degrees in rule-based expert
systems. Some of the methods proposed in the literature are briefly analysed
with particular attention to the Subjective Bayesian Probability (used in PROSPECTOR)
and the Confirmation Theory adopted in MYCIN.

The paper
presents an integrated approach based on Possibility Theory for evaluating the
degree of match between the set of conditions occurring in the antecedent of a
production rule and the input data, for combining the evidence degree of a fact
with the strength of implication of a rule and for combining evidence degrees
coming from different pieces of knowledge. The semantics of the logical
operators AND and OR in possibility theory and in our approach are compared.
Finally, the definitions of some quantifiers like AT LEAST *n*, AT
MOST *n*, EXACTLY *n* are introduced.

Cited by __62____ __

applications – document retrieval

M Lalmas, I Ruthven - Journal of Documentation 54, No 5, 1998

http://www.emeraldinsight.com/doi/abs/10.1108/EUM0000000007180

In this paper we report on a theoretical model of structured
document indexing and retrieval based on the Dempster‐Shafer Theory of Evidence. This
includes a description of our model of structured document retrieval, the
representation of structured documents, the representation of individual
components, how components are combined, details of the combination process,
and how relevance is captured within the model. We also present a detailed
account of an implementation of the model, and an evaluation scheme designed to
test the effectiveness of our model. Finally we report on the details and
results of a series of experiments performed to investigate the characteristics
of the model.

geometry – survey - distance

+ **Distances in evidence theory: Comprehensive survey and generalizations**

AL Jousselme, P Maupin - International Journal of Approximate Reasoning, Volume 53, Issue
2, February 2012, Pages 118–145

http://www.sciencedirect.com/science/article/pii/S0888613X1100106X

The purpose of the present work is to survey the
dissimilarity measures defined so far in the mathematical framework of evidence
theory, and to propose a classification of these measures based on their formal
properties. This research is motivated by the fact that while dissimilarity
measures have been widely studied and surveyed in the fields of probability
theory and fuzzy set theory, no comprehensive survey is yet available for
evidence theory. The main results presented herein include a synthesis of the
properties of the measures defined so far in the scientific literature; the
generalizations proposed naturally lead to additions to the body of the
previously known measures, leading to the definition of numerous new measures.
Building on this analysis, we have highlighted the fact that Dempster’s
conflict cannot be considered as a genuine dissimilarity measure between two
belief functions and have proposed an alternative based on a cosine function.
Other original results include the justification of the use of two-dimensional
indexes as (cosine; distance) couples and a general formulation for this class
of new indexes. We base our exposition on a geometrical interpretation of
evidence theory and show that most of the dissimilarity measures so far published
are based on inner products, in some cases degenerated. Experimental results
based on Monte Carlo simulations illustrate interesting relationships between
existing measures.

Cited by __76____ __

applications – accounting

**[book] Applications
of fuzzy sets and the theory of
evidence to accounting**

PH Siegel, A De Korvin, K Omer (editors) – Jai Press, 1995 - ISBN:
9780762304172

http://books.google.co.uk/books/about/Applications_of_fuzzy_sets_and_the_theor.html?id=SmpaAAAAYAAJ&redir_esc=y

An
analysis of fuzzy sets and the theory of evidence to accounting. It is divided
into parts, covering: methodology; inference; prediction; and neural networks.

other theories - possibility

- **On the uniqueness of possibilistic measure of uncertainty and
information**

GJ Klir, M Mariano - Fuzzy Sets and Systems 24, Issue 2, November 1987,
Pages 197–219

http://www.sciencedirect.com/science/article/pii/016501148790090X

It is demonstrated, through a series of theorems,
that the U-uncertainty (introduced by Higashi and Klir in 1982) is the only
possibilistic measure of uncertainty and information that satisfies
possibilistic counterparts of axioms of the well established Shannon and
hartley measures of uncertainty and information. Two complementary forms of the
possibilistic counterparts of the probabilistic branching (or grouping) axiom,
which is usually used in proofs of the uniqueness of the Shannon measure, are
introduced in this paper for the first time. A one-to-one correspondence between
possibility distributions and basic probabilistic assignments (introduced by
Shafer in his mathematical theory of evidence) is instrumental in most proofs
in this paper. The uniqueness proof is based on possibilistic formulations of
axioms of symmetry, expansibility, additivity, branching, monotonicity, and
normalization.

advances - arithmetics

**Arithmetic and other operations on Dempster-Shafer structures**

RR Yager - International Journal of Man-Machine Studies, Volume 25, Issue
4, October 1986, Pages 357–366

http://www.sciencedirect.com/science/article/pii/S0020737386800669

We show how variables whose values are represented
by Dempster-Shafer structures can be combined under arithmetic operations such
as addition. We then generalize this procedure to allow for the combination of
these types of variables under more general operations. We note that Dempster's
rule is a special case of this situation under the intersection operation.

other theories - GTU

+ **Toward a generalized theory of uncertainty (GTU) – an outline**

LA Zadeh - Information Sciences 172, Issues 1–2, 9 June 2005, Pages 1–40

http://www.sciencedirect.com/science/article/pii/S002002550500054X

It is a deep-seated tradition in science to view
uncertainty as a province of probability theory. The generalized theory of
uncertainty (GTU) which is outlined in this paper breaks with this tradition
and views uncertainty in a much broader perspective. Uncertainty is an
attribute of information. A fundamental premise of GTU is that information,
whatever its form, may be represented as what is called a generalized
constraint. The concept of a generalized constraint is the centerpiece of GTU.
In GTU, a probabilistic constraint is viewed as a special––albeit
important––instance of a generalized constraint. A generalized constraint is a
constraint of the form X isr R, where X is the constrained variable, R is a
constraining relation, generally non-bivalent, and r is an indexing variable
which identifies the modality of the constraint, that is, its semantics. The
principal constraints are: possibilistic (r = blank); probabilistic (r = p);
veristic (r = v); usuality (r = u); random set (r = rs); fuzzy graph (r = fg);
bimodal (r = bm); and group (r = g). Generalized constraints may be qualified,
combined and propagated. The set of all generalized constraints together with
rules governing qualification, combination and propagation constitutes the
generalized constraint language (GCL).

The generalized constraint language plays a key
role in GTU by serving as a precisiation language for propositions, commands
and questions expressed in a natural language. Thus, in GTU the meaning of a
proposition drawn from a natural language is expressed as a generalized constraint.
Furthermore, a proposition plays the role of a carrier of information. This is
the basis for equating information to a generalized constraint.

In GTU, reasoning under uncertainty is treated as
propagation of generalized constraints, in the sense that rules of deduction
are equated to rules which govern propagation of generalized constraints. A
concept which plays a key role in deduction is that of a protoform
(abbreviation of prototypical form). Basically, a protoform is an abstracted
summary––a summary which serves to identify the deep semantic structure of the
object to which it applies. A deduction rule has two parts: symbolic––expressed
in terms of protoforms––and computational.

GTU represents a significant change both in
perspective and direction in dealing with uncertainty and information. The
concepts and techniques introduced in this paper are illustrated by a number of
examples.

Cited by __827__

survey - foundations

J Kohlas, PA Monney - Zeitschrift für Operations Research 1994, Volume 39, Issue
1, pp 35-68

http://link.springer.com/article/10.1007%2FBF01440734

The mathematical theory of evidence has been
introduced by Glenn Shafer in 1976 as a new approach to the representation of
uncertainty. This theory can be represented under several distinct but more or
less equivalent forms. Probabilistic interpretations of evidence theory have
their roots in Arthur Dempster's multivalued mappings of probability spaces.
This leads to random set and more generally to random filter models of
evidence. In this probabilistic view evidence is seen as more or less probable
arguments for certain hypotheses and they can be used to support those
hypotheses to certain degrees. These degrees of support are in fact the
reliabilities with which the hypotheses can be derived from the evidence.
Alternatively, the mathematical theory of evidence can be founded axiomatically
on the notion of belief functions or on the allocation of belief masses to
subsets of a frame of discernment. These approaches aim to present evidence
theory as an extension of probability theory. Evidence theory has been used to
represent uncertainty in expert systems, especially in the domain of diagnostics.
It can be applied to decision analysis and it gives a new perspective for
statistical analysis. Among its further applications are image processing,
project planning and scheduling and risk analysis. The computational problems
of evidence theory are well understood and even though the problem is complex,
efficient methods are available.

debate - information

**Measures
of uncertainty in the Dempster-Shafer theory of evidence**

GJ Klir - Advances in the Dempster-Shafer theory of evidence, pages 35–49, Wiley and Sons, 1994 -
ISBN:0-471-55248-8

http://dl.acm.org/citation.cfm?id=186967

advances – independence - combination

**Representation,
independence, and combination of evidence in the Dempster-Shafer theory**

L Zhang - Advances in the
Dempster-Shafer theory of evidence,
pages 51–69, Wiley and Sons, 1994 ISBN:0-471-55248-8

http://repository.ust.hk/ir/Record/1783.1-41852

no abstract

Cited by __92__

machine learning - classification

+ **Handling possibilistic labels in pattern classification using evidential
reasoning**

T Denśux, LM Zouhal - Fuzzy Sets and Systems 122, Issue 3, 16 September
2001, Pages 409–424

http://www.sciencedirect.com/science/article/pii/S0165011400000865

A category of learning problems in which the class
membership of training patterns is assessed by an expert and encoded in the
form of a possibility distribution is considered. Each example i thus consists
in a feature vector xi and a possibilistic label , where denotes the possibility of that example
belonging to class k. This problem is tackled in the framework of Evidence
Theory. The evidential distance-based classifier previously introduced by one
of the authors is extended to handle possibilistic labeling of training data.
Two approaches are proposed, based either on the transformation of each
possibility distribution into a consonant belief function, or on the use of
generalized belief structures with fuzzy focal elements. In each case, a belief
function modeling the expert's beliefs concerning the class membership of each
new pattern is obtained. This information may then be either interpreted by a
human operator to support decision-making, or automatically processed to yield
a final class assignment through the computation of pignistic probabilities.
Experiments with synthetic and real data demonstrate the ability of both
classification schemes to make effective use of possibilistic labels as
training information.

advances - conditioning

+ **Jeffrey-like rules of conditioning for the Dempster-Shafer theory of evidence**

H Ichihashi, H Tanaka - International Journal of Approximate Reasoning, Volume 3, Issue 2, March
1989, Pages 143–156

http://www.sciencedirect.com/science/article/pii/0888613X89900030

Jeffrey's rule of conditioning is a rule for
changing an additive probability distribution when the human perception of new
evidence is obtained. It is a generalization of the normative Bayesian
inference. Shafer showed how Jeffrey's generalization of Bayes' rule of
conditioning can be reinterpreted in terms of the theory of belief functions.
But Shafer's approach is different from the normative Bayesian approach and is
not a straight generalization of Jeffrey's rule. There are situations in which
we need inference rules that may well provide a convenient generalization of
Jeffrey's rule. Therefore we propose new rules of conditioning motivated by the
work of Dubois and Prade. Although the weak and strong conditioning rules of
Dubois and Prade are generalizations of Bayesian conditioning, they fail to
yield Jeffrey's rule as a special case. Jeffrey's rule is a direct consequence
of a special case of our conditioning rules. Three kinds of normalizations in
the rules of conditioning are discussed.

applications – documents – information retrieval

+ **Using Dempster-Shafer's theory
of evidence to combine aspects of information use**

I Ruthven, M Lalmas - Journal of Intelligent Information Systems, Volume 19, Issue 3, pp 267-301, 2002

In this paper we propose a model for relevance
feedback. Our model combines evidence from user's relevance assessments with
algorithms describing how words are used within documents. We motivate the use
of the Dempster-Shafer framework as an appropriate theory for modelling
combination of evidence. This model also incorporates the uncertain nature of
information retrieval and relevance feedback. We discuss the sources of
uncertainty in combining evidence in information retrievel and the importance
of combining evidence in relevance feedback. We also present results from a
series of experiments that highlight various aspects of our approach and
discuss our findings.

Other theories - fuzzy

**k-order additive discrete fuzzy measures and their representation**

M Grabisch - Fuzzy Sets and Systems 92, Issue 2, 1 December 1997, Pages
167–189

http://www.sciencedirect.com/science/article/pii/S0165011497001681

In order to face with the complexity of discrete
fuzzy measures, we propose the concept of *k*-orderadditive fuzzy
measure, including usual additive measures and fuzzy measures. Every discrete
fuzzy measure is a*k*-order additive fuzzy measure for a unique *k*.
A related topic of the paper is to introduce an alternative representation of
fuzzy measures, called the interaction representation, which sets and extends
in a common framework the Shapley value and the interaction index proposed by
Murofushi and Soneda.

applications – quality control

+ **Detecting changes of steady states using the mathematical theory of evidence**

S Narasimhan, CS Kao, RSH Mah - AIChE journal, Volume 33, Issue 11, pages 1930-1932, 1987

http://onlinelibrary.wiley.com/doi/10.1002/aic.690331125/abstract

The detection of changes in steady states is
important in quality control, data reconciliation, and process monitoring
applications. In a previous paper (Narasimhan et al., 1986) a composite
statistical test was developed and evaluated for this purpose. In this note we
present and evaluate an alternative method based on the mathematical theory of
evidence developed by Shafer (1976). A key element of this approach is the
assignment of beliefs to the different propositions of interest. This step
usually involves subjective judgment. In this note we propose to make it less
subjective by using the probability distribution of the measurements and
certain limiting conditions. Simulation studies were carried out to compare the
performance of this method with the multivariate statistical test developed
by Narasimhan et al. (1986). The results
show that both methods give the same performance. Thus this approach is an
attractive alternative for detecting changes of steady states when the
variables are independent. The belief function proposed here may also be used
in other applications such as fault diagnosis (Kramer, 1987).

debate

**Classic works of the Dempster-Shafer theory of belief functions**

RR Yager, L Liu (Editors) Studies in Fuzziness and Soft Computing, Volume 219
2008, Springer ISBN: 978-3-540-25381-5

http://www.springer.com/gp/book/9783540253815

In other important
respects, however, the theory has not been moving forward. We still hear
questions that were asked in the 1980s: How do we tell if bodies of evidence
are independent? What do we do if they are dependent? We still encounter
confusion and disagreement about how to interpret the theory. And we still find
little acceptance of the theory in mathematical statistics, where it first
began 40 years ago.

We have come to
believe that three things are needed to move the theory forward.

• A richer
understanding of the uses of probability. Some authors, including our departed
friend Philippe Smets [6], have tried to distance the Dempster-Shafer theory
from the notion of probability. But we have long believed that the theory is
best regarded as a way of using probability [2, 4, 5]. Understanding of this
point is blocked by superficial but well entrenched dogmas that still need to
be overcome.

• A richer
understanding of statistical modeling. Mathematical statisticians and research
workers in many other communities have become accustomed to beginning an
analysis by specifying probabilities that are supposed known except for certain
parameters. Dempster-Shafer modelling uses a different formal starting point,
which may often be equally or more legitimate as a representation of actual
knowledge [3].

• Good examples.
The elementary introductions to the Dempster-Shafer theory that one finds in so
many different domains are inadequate guides for dealing with the complications
that arise in real problems. We need in-depth examples of sensible
Dempster-Shafer analyses of a variety ofproblems of real scientific and
technological importance.

other theories – capacities – moebius - geometry

A Chateauneuf, JY Jaffray - Mathematical Social Sciences 17, Issue 3, June
1989, Pages 263–283 - DOI: 10.1016/0165-4896(89)90056-5

http://www.sciencedirect.com/science/article/pii/0165489689900565

Monotone capacities (on finite sets) of finite or
infinite order (lower probabilities) are characterized by properties of their
Möbius inverses. A necessary property of probabilities dominating a given
capacity is demonstrated through the use of Gale's theorem for the
transshipment problem. This property is shown to be also sufficient if and only
if the capacity is monotone of infinite order. A characterization of dominating
probabilities specific to capacities of order 2 is also proved.

Cited by __405____ __

conflict

+ **Conflict management in Dempster–Shafer theory using the degree of
falsity**

J Schubert - International Journal of Approximate Reasoning 52, Issue 3, March
2011, Pages 449–460

http://www.sciencedirect.com/science/article/pii/S0888613X10001398

In this article we develop a method for conflict
management within Dempster–Shafer theory. The idea is that each piece of
evidence is discounted in proportion to the degree that it contributes to the
conflict. This way the contributors of conflict are managed on a case-by-case
basis in relation to the problem they cause. Discounting is performed in a
sequence of incremental steps, with conflict updated at each step, until the
overall conflict is brought down exactly to a predefined acceptable level.

applications – remote sensing - image

**Exemplifying the theory of
evidence in remote sensing image classification**

P Mertikas, ME Zervakis - International Journal of Remote Sensing, Volume 22, Issue 6,
2001

http://www.tandfonline.com/doi/abs/10.1080/01431160118597?journalCode=tres20#.VRLPJ_msV7g

This article introduces the mathematical theory of
evidence in classifying remote sensing images. Its main intent is to introduce
the less familiar concepts of belief functions in image classification. The
belief function can be considered as a generalisation of the classical Bayes
probability function that includes, however, a way to assess the strength of
evidence. To illustrate the theory of evidence, seven examples are given.

debate – belief update

+ **Evidence, knowledge, and belief functions**

D Dubois, H Prade - International Journal of Approximate Reasoning, Volume 6, Issue
3, May 1992, Pages 295–319

http://www.sciencedirect.com/science/article/pii/0888613X9290027W

This article tries to clarify some aspects of the
theory of belief functions especially with regard to its relevance as a model
for incomplete knowledge. It is pointed out that the mathematical model of
belief functions can be useful beyond a theory of evidence, for the purpose of
handling imperfect statistical knowledge. Dempster's rule of conditioning is
carefully examined and compared to upper and lower conditional probabilities.
Although both notions are extensions of conditioning, they cannot serve the
same purpose. The notion of focusing, as a change of reference class, is
introduced and opposed to updating. Dempster's rule is good for updating,
whereas the other form of conditioning expresses a focusing operation. In
particular, the concept of focusing models the meaning of uncertain statements
in a more natural way than updating. Finally, it is suggested that Dempster's
rules of conditioning and combination can be justified by the Bayers rule
itself. On the whole this article addresses most of the questions raised by
Pearl in the 1990 special issue of the International Journal of Approximate
Reasoning on belief functions and belief maintenance in artificial
intelligence.

other theories - fuzzy

+ **On the normalization of fuzzy belief structures**

RR Yager - International Journal of Approximate Reasoning, Volume 14,
Issues 2–3, February–April 1996, Pages 127–153

http://www.sciencedirect.com/science/article/pii/0888613X96000928

The issue of normalization in the fuzzy
Dempster-Shafer theory of evidence is investigated. We suggest a normalization
procedure called smooth normalization. It is shown that this procedure is a
generalization of the usual Dempster normalization procedure. We also show that
the usual process of normalizing an individual subnormal fuzzy subset by
proportionally increasing the membership grades until the maximum membership
grade is one is a special case of this smooth normalization process and in turn
closely related to the Dempster normalization process. We look an alternative
normalization process in the fuzzy Dempster-Shafer environment based on adding
to the membership grade of subnormal focal elements the amount by which the
fuzzy subset is subnormal.

other theories - fuzzy

**A general approach to parameter evaluation in fuzzy digital pictures**

D Dubois, MC Jaulent - Pattern Recognition Letters, Volume 6, Issue 4, September
1987, Pages 251–259

http://www.sciencedirect.com/science/article/pii/0167865587900857

A general approach to the evaluation of parameters
from fuzzy regions is outlined. The main idea is to consider a fuzzy subset of
an images as the nested family of its level-cuts, interpret this family as a
body of evidence in the sense of Shafer. Any intrinsic parameter can then be
calculated as a mathematical expectation based on a probability density
function. Fuzzy-valued parameters can also be derived. The approach encompasses
recent proposals by Rosenfeld for specific parameters such as perimeter,
diameter, etc., as well as the cardinality of a fuzzy set. It is also extended
to relational parameters between fuzzy regions in the image.

applications – sensor fusion – signal processing

+ **Multisensor signal processing in the framework of the theory of evidence**

A Appriou – Technical report ONERA-TP--99-196, 1999

In most of the requirements met in
situation assessment, multisensor analysis has to be able to recognize in
adverse conditions one situation out of a set of possibilities concerning for
instance either localization, identity, or matching hypotheses. To do so, it
uses measurements of more or less doubtful origin and prior knowledge that is
understood to be often poorly defmed, and whose validity is moreover difficult
to evaluate under real observation conditions. The present synthesis proposes a
generic modeling of this type of information in the framework of the theory of
evidence, with closer attention being paid to the different natures of data
processed in common cases. This modeling in then used to elaborate processing
methods able to face specific problems that may arise when multisensor systems
are implemented to achieve functions like detection, classification, matching
of ambiguous observations, or tracking. Crucial practical problems are more
specifically dealt with, such as appropriate combination processing and
decision making, management of heterogeneous frames of discernment, and
integration of contextual knowledge.

Furthermore, the interest of a global
functional approach at low level, possible in that framework, is emphasized.

applications – sensor fusion - aerospace

**Methods for multisensor classification of airborne targets integrating
evidence theory**

A Bastičre - Aerospace Science and Technology, Volume 2, Issue 6,
September 1998, Pages 401–411

http://www.sciencedirect.com/science/article/pii/S1270963899800285

This paper proposes to analyze methods applied to
the multisensor classification of airborne targets and which present the common
feature of using the theory of evidence developed by Dempster and Shafer. After
briefly outlining this technique, we deal more especially with the following
three methods: the global method proposed by G. Shafer, the separable method
recommended by A. Appriou and finally the extension of the standard *K* nearest
neighbors method proposed by T. Denoeux. These latter are particularly
appropriate in the treatment of multisensor classification problems since they
make it possible to consider non-exclusive hypotheses and to be able to
manipulate uncertain data. Several simulations relating to an airborne target
classification problem are presented. They demonstrate the advantages and
drawbacks of the various methods proposed and allow their respective behavior
to be studied. The early results obtained show that these methods are
particularly robust and perform well provided that certain hypotheses are
satisfied. A further avenue of research may consist of validating them on the
basis of actual measurements recorded using several different sensors.

other theories – frameworks – probabilistic logic

+ **A logic for reasoning about probabilities**

R Fagin, JY Halpern, N Megiddo - Information and computation, Volume 87, Issues 1–2,
July–August 1990, Pages 78–128

http://www.sciencedirect.com/science/article/pii/089054019090060U

We consider a language for reasoning about
probability which allows us to make statements such as “the probability
of *E*_{1} is less than 1/3” and “the probability
of *E*_{1} is at least twice the probability of *E*_{2},”
where *E*_{1}and *E*_{2} are arbitrary
events. We consider the case where all events are measurable (i.e., represent
measurable sets) and the more general case, which is also of interest in
practice, where they may not be measurable. The measurable case is essentially
a formalization of (the propositional fragment of) Nilsson's probabilistic
logic. As we show elsewhere, the general (nonmeasurable) case corresponds
precisely to replacing probability measures by Dempster-Shafer belief
functions. In both cases, we provide a complete axiomatization and show that
the problem of deciding satisfiability is NP-complete, no worse than that of
propositional logic. As a tool for proving our complete axiomatizations, we
give a complete axiomatization for reasoning about Boolean combinations of
linear inequalities, which is of independent interest. This proof and others
make crucial use of results from the theory of linear programming. We then extend
the language to allow reasoning about conditional probability and show that the
resulting logic is decidable and completely axiomatizable, by making use of the
theory of real closed fields.

Other theories – inference

+ **Foundations of probabilistic inference with uncertain evidence**

FJ Groen, A Mosleh - International Journal of Approximate Reasoning 39,
Issue 1, April 2005, Pages 49–83

http://www.sciencedirect.com/science/article/pii/S0888613X04000830

The
application of formal inference procedures, such as Bayes Theorem, requires
that a judgment is made, by which the evidential meaning of physical
observations is stated within the context of a formal model. Uncertain evidence
is defined as the class of observations for which this statement cannot take
place in certain terms. It is a significant class of evidence, since it cannot
be treated using Bayes Theorem in its conventional form [G. Shafer, A
Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ,
1976].

In this
paper, we present an extension of the Bayesian theory that can be used to perform
probabilistic inference with uncertain evidence. The extension is based on an
idealized view of inference in which observations are used to rule out possible
valuations of the variables in a modeling space.

The extension
is different from earlier probabilistic approaches such as Jeffrey’s rule of
probability kinematics and Cheeseman’s rule of distributed meaning, by
introducing two forms of evidential meaning representation are presented, for
which non-probabilistic analogues are found in theories such as Evidence Theory
and Possibility Theory. By viewing the statement of evidential meaning as a
separate step in the inference process, a clear probabilistic interpretation
can be given to these forms of representation, and a generalization of Bayes Theorem
can be derived. This generalized rule of inference allows uncertain evidence to
be incorporated into probabilistic inference procedures.

applications - medical

+ **A framework for intelligent medical diagnosis using the theory of evidence**

RW Jones, A Lowe, MJ Harrison - Knowledge-Based Systems, Volume 15,
Issues 1–2, January 2002, Pages 77–84

http://www.ingentaconnect.com/content/els/09507051/2002/00000015/00000001/art00123

In designing fuzzy logic systems for fault
diagnosis, problems can be encountered in the choice of symptoms to use fuzzy
operators and an inability to convey the reliability of the diagnosis using
just one degree of membership for the conclusion. By turning to an evidential
framework, these problems can be resolved whilst still preserving a fuzzy
relational model structure. The theory of evidence allows for utilisation of
all available information. Relationships between sources of evidence determine
appropriate combination rules. By generating belief and plausibility measures
it also communicates the reliability of the diagnosis, and completeness of
information. In this contribution medical diagnosis is considered using the
theory of evidence, in particular the diagnosis of inadequate analgesia is
considered.

survey – uncertainty measure

+ **Uncertainty measures for evidential reasoning I: A review**

NR Pal, JC Bezdek, R Hemasinha - International Journal of Approximate Reasoning 7, Issues
3–4, October–November 1992, Pages 165–183

http://www.sciencedirect.com/science/article/pii/0888613X9290009O

This paper is divided into two parts. Part I
discusses limitations of the measures of global uncertainty of Lamata and Moral
and total uncertainty of Klir and Ramer. We prove several properties of
different nonspecificity measures. The computational complexity of different
total uncertainty measures is discussed. The need for a new measure of total
uncertainty is established in Part I. In Part II, we propose a set of
intuitively desirable axioms for a measure of total uncertainty and then derive
an expression for the same. Several theorems are proved about the new measure.
The proposed measure is additive, and unlike other measures, has a unique
maximum. This new measure reduces to Shannon's probabilistic entropy when the
basic probability assignment focuses only on singletons. On the other hand,
complete ignorance—basic assignment focusing only on the entire set, as a whole—reduces
it to Hartley's measure of information. The computational complexity of the
proposed measure is O(N), whereas the previous measures are O(N^{2}).

Cited by __71__

applications – measurement – random fuzzy variables

A Ferrero, S Salicone - IEEE Transactions on Instrumentation and Measurement, Volume
56, Issue 3, pages 704 – 716, 2007 - DOI:10.1109/TIM.2007.894907

Random-fuzzy variables (RFVs) are mathematical
variables defined within the theory of evidence. Their importance in
measurement activities is due to the fact that they can be employed for the
representation of measurement results, together with the associated
uncertainty, whether its nature is random effects, systematic effects, or unknown
effects. Of course, their importance and usability also depend on the fact that
they can be employed for processing measurement results. This paper proposes
suitable mathematics and related calculus for processing RFVs, which consider
the different nature and the different behavior of the uncertainty effects. The
proposed approach yields to process measurement algorithms directly in terms of
RFVs so that the final measurement result (and all associated available
information) is provided as an RFV.

Applications – ubiquitous computing

+ **Using Dempster-Shafer theory
of evidence for situation inference**

S McKeever, J Ye, L Coyle, S Dobson - Smart Sensing and Context, Lecture Notes in Computer Science
Volume 5741, 2009, pp 149-162

In the domain of ubiquitous computing, the ability
to identify the occurrence of situations is a core function of being
’context-aware’. Given the uncertain nature of sensor information and inference
rules, reasoning techniques that cater for uncertainty hold promise for
enabling the inference process. In our work, we apply the Dempster Shafer
theory of evidence to infer situation occurrence with minimal use of training
data. We describe a set of evidential operations for sensor mass functions
using context quality and evidence accumulation for continuous situation
detection. We demonstrate how our approach enables situation inference with
uncertain information using a case study based on a published smart home
activity data set.

applications – classification – remote sensing – earth sciences

+ **Classification of a complex landscape using Dempster–Shafer theory of evidence**

L Cayuela, JD Golicher, JS Rey and JM Rey Benajas - International Journal of Remote Sensing,
Volume 27, Issue 10, pages 1951-1971, 2006

http://www.tandfonline.com/doi/pdf/10.1080/01431160500181788

The landscape of the Highlands of Chiapas, southern
Mexico, is covered by a highly complex mosaic of anthropogenic, natural and
semi‐natural vegetation. This complexity challenges land cover
classification based on remotely sensed data alone. Spectral signatures do not
always provide the basis for an unambiguous separation of pixels into classes.
Expert knowledge does, however, provide additional lines of evidence that can
be employed to modify the belief that a pixel belongs to a certain coverage
class. We used Dempster–Shafer (DS) weight of evidence modelling to incorporate
this information into the classification process in a formal manner. Expert knowledge‐based
variables were related to: (1) altitude, (2) slope, (3) distance to known human
settlements and (4) landscape perceptions regarding dominance of vegetation
types. The results showed an improvement of classification results compared with
traditional classifiers (maximum likelihood) and context operators (modal
filters), leading to better discrimination between categories and (i) a
decrease in errors of omission and commission for almost all classes and (ii) a
decrease in total error of around 7.5%. The DS approach led not only to a more
accurate classification but also to a richer description of the inherent
uncertainty surrounding it.

debate

+ **Epistemic
logics, probability, and the calculus of evidence**

EH Ruspini - Proceedings of the 10^{th} International Joint
Conference on Artificial Intelligence (IJCAI'87), Vol 2, pages 924-931, 1987

This paper presents results of the application to
epistemic logic structures of the method proposed by Carnap for the development
of logical foundations of probability theory. These results, which provide firm
conceptual bases for the Dempster-Shafer calculus of evidence, are derived by
exclusively using basic concepts from probability and modal logic theories,
without resorting to any other theoretical notions or structures. A form of
epistemic logic (equivalent in power to the modal system S5), is used to define
a space of possible worlds or states of affairs. This space, called the
epistemic universe, consists of all possible combined descriptions of the state
of the real world and of the state of knowledge that certain rational agents have
about it. These representations generalize those derived by Carnap, which were
confined exclusively to descriptions of possible states of the real world.

Probabilities defined on certain classes of sets of
this universe, representing different states of knowledge about the world, have
the properties of the major functions of the Dempster-Shafer calculus of
evidence: belief functions and mass assignments. The importance of these
epistemic probabilities lies in their ability to represent the effect of uncertain
evidence in the states of knowledge of rational agents. Furthermore, if an
epistemic probability is extended to a probability function defined over
subsets of the epistemic universe that represent true states of the real world,
then any such extension must satisfy the well-known interval bounds derived
from the Dempster-Shafer theory.

Application of this logic-based approach to
problems of knowledge integration results in a general expression, called the
additive combination formula, which can be applied to a wide variety of
problems of integration of dependent and independent knowledge. Under
assumptions of probabilistic independence this formula is equivalent to
Dempster's rule of combination.

applications - communications

N Nguyen-Thanh, I Koo - IEEE Communications Letters, Volume 13, Issue 7, pages 492 –
494, July 2009

This letter proposes an enhanced scheme for
cooperative spectrum sensing which utilizes the signal to noise ratios to
evaluate the degree of reliability of each local spectrum sensing terminal on a
distributed Cognitive Radio network. The terminals' reliability weight is applied
to adjust its sensing data more accurately before making fusion by
Dempter-Shafer theory of evidence. Simulation results show that significant
improvement of the cooperative spectrum sensing gain is achieved by our scheme.

Cited by __53__

other theories – rough sets

+ **Vagueness and uncertainty: a rough set perspective**

Z Pawlak - Computational Intelligence, Volume 11, Issue 2, pages
227–232, May 1995

http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8640.1995.tb00029.x/abstract

Vagueness and uncertainty have attracted the
attention of philosophers and logicians for many years. Recently, AI
researchers contributed essentially to this area of research. Fuzzy set theory
and the theory of evidence are seemingly the most appealing topics. On this
note we present a new approach, based on the rough set theory, for looking to
these problems. The theory of rough sets seems a suitable mathematical tool for
dealing with problems of vagueness and uncertainty. This paper is a modified
version of the author's lecture titled “An inquiry into vagueness and
uncertainty,” which was delivered at the AI Conference in Wigry (Poland), 1994.

applications - engineering

**Investigation
of evidence theory for engineering applications**

WL Oberkampf, JC Helton - AIAA Non-Deterministic Approaches Forum, 2002

http://arc.aiaa.org/doi/abs/10.2514/6.2002-1569

no
abstract

Cited by __93__

classification – applications - speech

+ **Speaker identification by combining multiple classifiers using
Dempster–Shafer theory of evidence**

H Altınçay, M Demirekler - Speech Communication, Volume 41, Issue 4, November
2003, Pages 531–547

This paper presents a multiple classifier approach
as an alternative solution to the closed-set text-independent speaker
identification problem. The proposed algorithm which is based on Dempster–Shafer
theory of evidence computes the first and *R*th level ranking
statistics. *R*th level confusion matrices extracted from these
ranking statistics are used to cluster the speakers into model sets where they
share set specific properties. Some of these model sets are used to reflect the
strengths and weaknesses of the classifiers while some others carry speaker
dependent ranking statistics of the corresponding classifier. These information
sets from multiple classifiers are combined to arrive at a joint decision. For
the combination task, a rule-based algorithm is developed where Dempster’s rule
of combination is applied in the final step. Experimental results have shown
that the proposed method performed much better compared to some other
rank-based combination methods.

computation – approximation - combination

T Denśux, AB Yaghlane - International Journal of Approximate Reasoning, Volume
31, Issues 1–2, October 2002, Pages 77–101

http://www.sciencedirect.com/science/article/pii/S0888613X02000737

A method is proposed for reducing the size of a frame
of discernment, in such a way that the loss of information content in a set of
belief functions is minimized. This method may be seen as a hierarchical
clustering procedure applied to the columns of a binary data matrix, using a
particular dissimilarity measure. It allows to compute approximations of the
mass functions, which can be combined efficiently in the coarsened frame using
the fast Möbius transform algorithm, yielding inner and outer approximations of
the combined belief function.

applications – earth sciences

- **Application of Dempster-Shafer theory of evidence to GIS-based landslide susceptibility
analysis**

NW Park - Environmental Earth Sciences, Volume 62, Issue 2, pp 367-376,
2011

http://link.springer.com/article/10.1007%2Fs12665-010-0531-5

GIS-based spatial data integration tasks for
predictive geological applications, such as landslide susceptibility analysis,
have been regarded as one of the primary geological application issues of GIS.
An efficient framework for proper representation and integration is required
for this kind of application. This paper presents a data integration framework
based on the Dempster-Shafer theory of evidence for landslide susceptibility
mapping with multiple geospatial data. A data-driven information representation
approach based on spatial association between known landslide occurrences and
input geospatial data layers is used to assign mass functions. After defining
mass functions for multiple geospatial data layers, Dempster’s rule of
combination is applied to obtain a series of combined mass functions. Landslide
susceptibility mapping using multiple geospatial data sets from Jangheung in
Korea was conducted to illustrate the application of this methodology. The
results of the case study indicated that the proposed methodology efficiently
represented and integrated multiple data sets and showed better prediction
capability than that of a traditional logistic regression model.

debate – conditioning – belief update – credal sets - intervals

+ **Bayesian and non-Bayesian evidential updating**

HE Kyburg Jr - Artificial Intelligence, Volume 31, Issue 3, March 1987,
Pages 271–293

http://www.sciencedirect.com/science/article/pii/0004370287900683

Four
main results are arrived at in this paper. (1) Closed convex sets of classical
probability functions provide a representation of belief that includes the
representations provided by Shafer probability mass functions as a special
case. (2) The impact of “uncertain evidence” can be (formally) represented by Dempster conditioning, in
Shafer's framework. (3) The impact of “uncertain
evidence” can be (formally) represented in the
framework of convex sets of classical probabilities by classical
conditionalization. (4) The probability intervals that result from
Dempster-Shafer updating on uncertain evidence are included in (and may be
properly included in) the intervals that result from Bayesian updating on
uncertain evidence.

applications – civil engineering

R Sadiq, Y Kleiner, B Rajani - Civil Engineering and Environmental Systems,
Volume 23, Issue 3, pages 129-141, 2006

Intrusion of contaminants into water distribution
networks requires the simultaneous presence of three elements: contamination
source, pathway and driving force. The existence of each of these elements
provides ‘partial’ evidence (typically incomplete and non-specific) to the
occurrence of contaminant intrusion into distribution networks. Evidential reasoning,
also called Dempster–Shafer theory, has proved useful to incorporate both *aleatory* and *epistemic* uncertainties
in the inference mechanism. The application of evidential reasoning to assess
risk of contaminant intrusion is demonstrated with the help of an example of a
single pipe. The proposed approach can be extended to full-scale water distribution
networks to establish risk-contours of contaminant intrusion. Risk-contours
using GIS may help utilities to identify sensitive locations in the water
distribution network and prioritize control and preventive strategies.

decision making

+ **Reformulating decision theory using fuzzy set theory and Shafer's theory of evidence**

RF Bordley - Fuzzy Sets and Systems 139, Issue 2, 16 October 2003, Pages
243–266

http://www.sciencedirect.com/science/article/pii/S0165011402005158

Utilities and probabilities in
decision theory are usually assessed by asking individuals to indicate their
preferences between various uncertain choices. In this paper, we argue that

(1) The utility of a consequence can
be assessed as the membership function of the consequence in the fuzzy set ‘*satisfactory*’.

(2) The probability of an event,
instead of being directly assessed, should be inferred from the *evidence*associated
with that event. The degree of evidence is quantified using Shaferian basic
probability assignments.

In addition, we use the Heisenberg
Uncertainty Principle to argue for a change in one of the technical assumptions
underlying decision theory. As a result of this change, some kinds of evidence
will be observable in certain experiments but unobservable in others. Since
probabilities are defined over the potential outcomes of an experiment, they
will only be defined over some, but not all, the evidence. As a result, the
probabilities associated with different experiments could be inconsistent.

This formulation emphasizes the
importance of new distinctions (and not just new information) in updating
probabilities. We argue that this formulation addresses many of the observed
empirical deviations between decision theory and experiment. It also addresses
the anomalies of quantum physics. We close with a brief discussion of directions
for further research.

applications - medical

L Jones, MJ Beynon, CA Holt, S Roy - Journal of Biomechanics 39, Issue 13, 2006, Pages 2512–2520

http://www.sciencedirect.com/science/article/pii/S0021929005003635

This paper utilises a novel method for the
classification of subjects with osteoarthritic and normal knee function. The
classification method comprises a number of different components. Firstly, the
method exploits the Dempster–Shafer theory of evidence allowing for a degree of
ignorance in the subject's classification, i.e., a level of uncertainty as to
whether a gait variable indicates osteoarthritis or not. Secondly, the
inclusion of simplex plots allows both the classification of a subject, and the
contribution of each associated gait variable to that classification, to be
represented visually. As a result, the method is further able to highlight
periodic changes in a subject's knee function due to total knee replacement
surgery and subsequent recovery. The visual representation enables a simple
clinical interpretation of the results from the quantitative analysis.

combination

**Efficient combination rule of evidence theory**

B Li, B Wang, J Wei, Y Huang and Z Guo - Proc. SPIE 4554, Object
Detection, Classification, and Tracking Technologies, 237 (September 24, 2001);
doi:10.1117/12.441655

http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=899447

D-S evidence theory is a useful method in dealing
with uncertainty problems, but its application is limited because of the
shortcomings of its combination rule. This paper present an efficient
combination rule, that is, the evidences' conflicting probability is
distributed to every proposition according to its average supported degree. The
new combination rule improves the reliability and rationality of combination
results. Although evidences conflict one another highly, good combination
results are also obtained.

Cited by __51____ __

other theories - incidence

+ **On some equivalence relations
between incidence calculus and Dempster-Shafer theory of evidence**

FC da Silva, A Bundy - arXiv preprint arXiv:1304.1126, 2013

Incidence Calculus and Dempster-Shafer Theory of
Evidence are both theories to describe agents' degrees of belief in
propositions, thus being appropriate to represent uncertainty in reasoning
systems. This paper presents a straightforward equivalence proof between some
special cases of these theories.

applications - environment

MJ Ducey - Forest Ecology and Management 150, Issue 3, 15 September
2001, Pages 199–211

http://www.sciencedirect.com/science/article/pii/S037811270000565X

Forest management decisions often must be made
using sparse data and expert judgment. The representation of this knowledge in
traditional approaches to decision analysis implies a precise value for
probabilities or, in the case of Bayesian analysis, a precisely specified joint
distribution for unknown parameters. The precision of this specification does
not depend on the strength or weakness of the evidence on which it is based.
This often leads to exaggerated precision in the results of decision analyses,
and obscures the importance of imperfect information. Here, I suggest an
alternative based on the Dempster–Shafer theory of evidence, which differs from
conventional approaches in allowing the allocation of belief to subsets of the
possible outcomes, or, in the case of a continuous set of possibilities, to
intervals. The Dempster–Shafer theory incorporates Bayesian analysis as a
special case; a critical difference lies in the representation of ignorance or
uncertainty. I present examples of silvicultural decision-making using belief
functions for the case of no data, sparse data, and adaptive management under
increasing data availability. An approach based on the Dempster–Shafer
principles can yield not only indications of optimal policies, but also
valuable information about the level of certainty in decision-making.

machine learning - classification – fusion – applications -
geoscience

L Hegarat-Mascle, I Bloch and D Vidal-Madjar - IEEE Transactions on
Geoscience and Remote Sensing 35, no. 4, July 1997

The aim of this paper is to show that
Dempster-Shafer evidence theory may be successfully applied to unsupervised
classification in multisource remote sensing. Dempster-Shafer formulation
allows for consideration of unions of classes, and to represent both
imprecision and uncertainty, through the definition of belief and plausibility
functions. These two functions, derived from mass function, are generally
chosen in a supervised way. In this paper, the authors describe an unsupervised
method, based on the comparison of monosource classification results, to select
the classes necessary for Dempster-Shafer evidence combination and to define
their mass functions. Data fusion is then performed, discarding invalid
clusters (e.g. corresponding to conflicting information) thank to an iterative
process. Unsupervised multisource classification algorithm is applied to
MAC-Europe'91 multisensor airborne campaign data collected over the Orgeval
French site. Classification results using different combinations of sensors
(TMS and AirSAR) or wavelengths (L- and C-bands) are compared. Performance of
data fusion is evaluated in terms of identification of land cover types. The
best results are obtained when all three data sets are used. Furthermore, some
other combinations of data are tried, and their ability to discriminate between
the different land cover types is quantified.

machine learning - classification

+ **Analysis of evidence-theoretic decision rules for pattern classification**

T Denoeux - Pattern Recognition 30, Issue 7, July 1997, Pages 1095–1107

http://www.sciencedirect.com/science/article/pii/S0031320396001379

The Dempster-Shafer theory provides a convenient
framework for decision making based on very limited or weak information. Such
situations typically arise in pattern recognition problems when patterns have
to be classified based on a small number of training vectors, or when the
training set does not contain samples from all classes. This paper examines
different strategies that can be applied in this context to reach a decision (e.g.
assignment to a class or rejection), provided the possible consequences of each
action can be quantified. The corresponding decision rules are analysed under
different assumptions concerning the completeness of the training set. These
approaches are then demonstrated using real data.

debate

**+ Review
of Mathematical theory of evidence,
by Glenn Shafer**

L Zadeh - AI Magazine, 1984

The
seminal work of Glenn Shafer – which is based on an earlier work by Arthur
Dempster – was published at a time when the theory of expert systems was in its
infancy and there was little interest within the AI community in issues
relating to probabilistic or evidential reasoning.

Cited by __445____ __

applications - retrieval

**- A
model of an information retrieval system based on situation theory and
Dempster-Shafer theory of evidence**

M Lalmas, CJ van Rijsbergen - Proceedings of the 1st Workshop on
Incompleteness and Uncertainty in Information Systems, Montreal, Canada (1993),
pp. 62–67

no
abstract

applications - water

R Sadiq, MJ Rodriguez - Chemosphere, Volume 59, Issue 2, April 2005, Pages 177–188

Interpreting water quality data routinely generated
for control and monitoring purposes in water distribution systems is a
complicated task for utility managers. In fact, data for diverse water quality
indicators (physico-chemical and microbiological) are generated at different
times and at different locations in the distribution system. To simplify and
improve the understanding and the interpretation of water quality,
methodologies for aggregation and fusion of data must be developed. In this
paper, the Dempster–Shafer theory also called theory of evidence is introduced
as a potential methodology for interpreting water quality data. The conceptual
basis of this methodology and the process for its implementation are presented
by two applications. The first application deals with the interpretation of
spatial water quality data fusion, while the second application deals with the
development of water quality index based on key monitored indicators. Based on
the obtained results, the authors discuss the potential contribution of theory
of evidence as a decision-making tool for water quality management.

applications - retrieval

M Lalmas, E Moutogianni – Proc. of the RIAO conference, Paris, France, 2000

Effective retrieval of hierarchically structured
web documents should exploit the content and structural knowledge associated
with the documents. This knowledge can be used to retrieve optimal documents:
documents that contain relevant information, and from which users can browse,
using the links in these documents, to retrieve further relevant documents. We
refer to this approach as focussed retrieval. This paper investigates the
effectiveness of a model for the focussed retrieval of hierarchically
structured web documents based on the Dempster-Shafer theory of evidence. To
allow for focussed retrieval, the representation of a document is defined as
the aggregation of the representation of its own content and that of its child
documents. To evaluate the model, we constructed a test collection based on a
museum web site. From our experiments on this collection, the results show that
the Dempster-Shafer theory, in particular, the aggregation, leads to an
effective focussed retrieval of hierarchically structured web documents.

approximations - decision

+ **Approximations
for decision making in the Dempster-Shafer theory of evidence**

M Bauer - Proceedings of the Twelfth International Conference on
Uncertainty in Artificial Intelligence (UAI’96), pages 73-80, 1996

The computational complexity of reasoning within
the Dempster-Shafer theory of evidence is one of the main points of criticism
this formalism has to face. To overcome this difficulty various approximation
algorithms have been suggested that aim at reducing the number of focal
elements in the belief functions involved, Besides introducing a new algorithm
using this method, this paper describes an empirical study that examines the
appropriateness of these approximation procedures in decision making
situations. It presents the empirical findings and discusses the various
tradeoffs that have to be taken into account when actually applying one of
these methods.

fusion

+ **Sensor fusion using Dempster-Shafer theory**

H Wu, M Siegel, R Stiefelhagen and J Yang - Proceedings of the 19th Instrumentation and Measurement
Technology Conference, Vol 1, pages 7-12, 2002

Context-sensing for context-aware HCI challenges
the traditional sensor fusion methods with dynamic sensor configuration and
measurement requirements commensurate with human perception. The
Dempster-Shafer theory of evidence has uncertainty management and inference
mechanisms analogous to our human reasoning process. Our Sensor Fusion for
Context-aware Computing Project aims to build a generalizable sensor fusion
architecture in a systematic way. This naturally leads us to choose the
Dempster-Shafer approach as our first sensor fusion implementation algorithm
This paper discusses the relationship between Dempster-Shafer theory and the
classical Bayesian method, describes our sensor fusion research work using
Dempster-Shafer theory in comparison with the weighted sum of probability
method The experimental approach is to track a user's focus of attention from
multiple cues. Our experiments show promising, thought-provoking results
encouraging further research.

applications - geoscience

H Kim, PH Swain - Geoscience and Remote Sensing Symposium, IGARSS'89, Vol 2,
1989

While it is empirically reasonable to assume that
multispectral data have the multivariate Gaussian distribution, geographic or
topographic data combined with multispectral data may not be represented by any
parametric model. Furthermore, there is a difficutly in describing the various
data types which have different units of measurements. These problems have been
the motivation for the development of classification techniques in which
various sources of data are assessed separately, and individual assessments are
combined by some means.

In this paper, we present a method for mutlisource
data classification based on the Shafer's mathematical theory of evidence. In
this method, data, sources are considered as entirely distinct bodies of
evidence providing subjective probabilistic measures to propositions. In order
to aggregate the information from multiple sources, the method adopts

Dempster's rule for combining multiple bodies of
evidence.
The focuses of the paper are on 1) construction of support functions (or
plausibility functions) given a body of statistical evidence, and 2) inference mechanisms of Dempster's
rule in combining information under uncertamty.

Preliminary experiments have been undertaken to illustrate
the use of
the method in a supervised ground-cover type classification on multispectral
data combined with digital elevation data. They demonstrate the ability of the
method in capturing information provided .by inexact and incomplete evidence
when there are not enough training samples to

estimate statistical parameters.

other theories - endorsements

+ **A theory of heuristic reasoning about uncertainty**

PR Cohen, MR Grinberg - AI magazine, Volume 4, Number 2, 1983

http://www.aaai.org/ojs/index.php/aimagazine/article/view/393

This article describes a theory of reasoning about
uncertainty, based on a representation of states of certainty called
endorsements. The theory of endorsements is an alternative to numerical methods
for reasoning about uncertainty, such as subjective Bayesian methods
(Shortliffe and Buchanan, 1975; Duda hart, and Nilsson, 1976) and Shafer-Dempster
theory (Shafer, 1976). The fundamental concern with numerical representations
of certainty is that they hide the reasoning about uncertainty. While numbers
are easy to propagate over inferences, what the numbers mean is unclear. The
theory of endorsements provide a richer representation of the factors that
affect certainty and supports multiple strategies for dealing with uncertainty.

other theories – perception-based

+ **Toward a perception-based theory of probabilistic reasoning with
imprecise probabilities**

LA Zadeh - Journal of Statistical Planning and Inference 105, Issue 1, 15
June 2002, Pages 233–264

http://link.springer.com/chapter/10.1007%2F978-3-7908-1773-7_2

The perception-based theory of probabilistic
reasoning which is outlined in this paper is not in the traditional spirit. Its
principal aim is to lay the groundwork for a radical enlargement of the role of
natural languages in probability theory and its applications, especially in the
realm of decision analysis. To this end, probability theory is generalized by
adding to the theory the capability to operate on perception-based information,
e.g., “Usually Robert returns from work at about 6 p.m.” or “It is very
unlikely that there will be a significant increase in the price of oil in the near
future”. A key idea on which perception-based theory is based is that the
meaning of a proposition, *p*, which describes a perception, may be
expressed as a generalized constraint of the form *X* is*rR*,
where *X* is the constrained variable, *R* is the
constraining relation and is*r* is a copula in which *r* is
a discrete variable whose value defines the way in which *R* constrains *X*.
In the theory, generalized constraints serve to define imprecise probabilities,
utilities and other constructs, and generalized constraint propagation is
employed as a mechanism for reasoning with imprecise probabilities as well as
for computation with perception-based information.

Cited by __310____ __

applications - communications

P Qihang, Z Kun, W Jun and L Shaoqian - IEEE 17th International
Symposium on Personal, indoor and Mobile Radio Communications, pages 1-5, 2006
- DOI:10.1109/PIMRC.2006.254365

honey_puma@yahoo.com.cn, and {zengkun, junwang, lsq}@uestc.edu.cn

Reliable detection of available spectrum is the
foundation of cognitive radio technology. To improve the detection probability
under sustainable false alarm rate, a distributed spectrum sensing scheme has
been proposed. In this paper, we propose a new decision combination scheme, in
which the credibility of local spectrum sensing is taken into account in the
final decision at central access point and Dempster-Shafer's evidence theory is
adopted to combine different sensing decisions from each cognitive user.
Simulation results show that significant improvement in detection probability
as well as reduction in false alarm rate is achieved by our proposal.

debate – survey - evidence

+ **Weight of
evidence: A brief survey**

IJ Good – Bayesian Statistics 2 (JM Bernardo, MH de Groot, DV Lindley
and AFM Smith editors), pages 249-270, 1985

A review
is given of the concepts of Bayes factors and weights of evidence, including
such aspects as terminology, uniqueness of the explicatum, history, how to make
judgements, and the relationship to tail-area probabilities. …

I have not
yet understood Shafer's theory of evidence, which is based on Dempster's
previous work on interval-valued probabilities. Aitchison (1968) seems to me to
have refuted the approach that Dempster mistake **...**

evidence

+ **On considerations of credibility of evidence**

RR Yager - International Journal of Approximate Reasoning 7, Issues 1–2,
August–September 1992, pp. 45–72

http://www.sciencedirect.com/science/article/pii/0888613X9290024T

The focus of this work is on the issue of managing
credibility information in reasoning systems. We first discuss a closely
related idea, that of multicriteria decision making. It is shown how the
concept of importance in multicriteria decision making is similar to the
concept of credibility in evidential reasoning systems. A new concept of
credibility qualification is introduced in both the theory of approximate
reasoning and the mathematical theory of evidence. A concept of relative
credibility is also introduced. This relative credibility is useful in
situations where the credibility of a piece of evidence is determined by its
compatibility with higher priority evidence.

Cited by __19____ __

conditioning - evidence

+ **Conditioning and updating evidence**

EC Kulasekere, K Premaratne, D.A. Dewasurendrab, M.-L. Shyub and P.H. Bauer - International Journal
of Approximate Reasoning 36, Issue 1, April 2004, Pages 75–108

http://www.sciencedirect.com/science/article/pii/S0888613X03001269

A new interpretation of Dempster–Shafer conditional
notions based *directly* upon the mass assignments is provided.
The masses of those propositions that *may* imply the complement
of the conditioning proposition are shown to be completely annulled by the
conditioning operation; conditioning may then be construed as a re-distribution
of the masses of some of these propositions to those that *definitely* imply
the conditioning proposition. A complete characterization of the propositions
whose masses are annulled without re-distribution, annulled with
re-distribution and enhanced by the re-distribution of masses is provided. A
new evidence updating strategy that is composed of a linear combination of the
available evidence and the conditional evidence is also proposed. It enables
one to account for the ‘integrity’ and ‘inertia’ of the available evidence and
its ‘flexibility’ to updating by appropriate selection of the linear
combination weights. Several such strategies, including one that has a
probabilistic interpretation, are also provided.

applications - engineering

**Assessment of structural damage using the theory of evidence**

S Toussi, JTP Yao - Structural Safety 1, Issue 2, 1982–1983, Pages
107–121

http://www.sciencedirect.com/science/article/pii/0167473082900194

An attempt has been made to analyze dynamic test
data of building structures for the assessment of structural damage. In this
paper, the method of converting evidential information to the interval
representation of Dempster and Shafer is applied. It is shown how results as
obtained from an individual source are interpreted. Then, the information as
obtained from several different sources is combined to compensate for individual
deficiencies of the knowledge sources. The resulting algorithm is then applied
to assess the damage state of a prototype 10-story reinforced concrete frame
structure subjected to repeated earthquake loading conditions in the
laboratory.

fusion – applications - image processing

- **Active fusion - A new method applied to remote sensing image
interpretation**

A Pinz, M Prantl, H Ganster and H Kopp-Borotschnig - Pattern Recognition Letters
Volume 17, Issue 13, 25 November 1996, Pages 1349–1359

http://www.sciencedirect.com/science/article/pii/S016786559600092X

Today's computer vision applications often have to
deal with multiple, uncertain and incomplete visual information. In this paper,
we introduce a new method, termed ‘*active fusion*’, which provides a
common framework for active selection and combination of information from
multiple sources in order to arrive at a reliable result at reasonable costs.
The implementation of active fusion on the basis of probability theory, the
Dempster-Shafer theory of evidence and fuzzy sets is discussed. In a sample
experiment, active fusion using Bayesian networks is applied to agricultural
field classification from multitemporal Landsat imagery. This experiment shows
a significant reduction of the number of information sources required for a
reliable decision.

applications - reliability

**Fault tree analysis in an early design stage using the
Dempster-Shafer theory of evidence**

P Limbourg, R Savic, J Petersen and HD Kochs - Risk, Reliability and
Societal Safety 2007 - ISBN 978-0-415-44786-7

The Dempster-Shafer Theory of
Evidence (DST) has been considered as an alternative to probabilistic modelling
if both a large amount of uncertainty and a conservative treatment of this
uncertainty are necessary. Both requirements are normally met in early design
stages. Expert estimates replace field data and hardly any accurate test
results are available. Therefore, a conservative uncertainty treatment is
beneficial to assure a reliable and safe design. The paper investigates the
applicability of DST which merges interval-based and probabilistic uncertainty
modelling on a fault tree analysis from the automotive area. The system under
investigation, an automatic transmission from the ZF AS Tronic series is still
in the development stage. We investigate the aggregation of expert estimates
and the propagation of the resulting mass function through the system model. An
exploratory sensitivity based on a nonspecifity measure indicates which
components contribute to the overall model uncertainty. The results are used to
predict if the system complies with a given target failure measure.

other theories
– expert systems

**+ [BOOK]**** ****Rule-based expert systems**__ - The
MYCIN Experiments of the Stanford Heuristic Programming Project__

BG Buchanan, EH Shortliffe – Addison–Wesley, 1984

Chapter 13 The Dempster-Shafer Theory of Evidence 272 Jean Gordon and
Edward H. Shortliffe

information

**Properties of measures of information in evidence and possibility
theories**

D Dubois, H Prade - Fuzzy Sets and Systems 100, Supplement 1, 1999, Pages 35–49

http://www.sciencedirect.com/science/article/pii/S0165011499800050

An overview of information measures recently
introduced by several authors in the settting of Shafer's theory of evidence is
proposed. New results pertaining to additivity and monotonicity properties of these
measures of information are presented. The interpretation of each measure of
information as opposed to others is discussed. The potential usefulness of
measures of specificity or imprecision is suggested, and a ‘principle of
minimal specificity’ is stated for the purpose of reconstructing a body of
evidence from incomplete knowledge.

other theories - hints

+ **Allocation of arguments and evidence theory**

J Kohlas - Theoretical Computer Science, Volume 171, Issues 1–2, 15 January
1997, Pages 221–246

http://www.sciencedirect.com/science/article/pii/S0304397596001302

The Dempster-Shafer theory of evidence is developed
here in a very general setting. First, its symbolic or algebraic part is
discussed as a body of arguments which contains an allocation of support and an
allowment of possibility for each hypothesis. It is shown how such bodies of
arguments arise in the theory of hints and in assumption-based reasoning in
logic. A rule of combination of bodies of arguments is then defined which
constitutes the symbolic counterpart of Dempster's rule. Bodies of evidence are
next introduced by assigning probabilities to arguments. This leads to support
and plausibility functions on some measurable hypotheses. As expected in
Dempster-Shafer theory, they are shown to be set functions, monotone or
alternating of infinite order, respectively. It is shown how these support and
plausibility functions can be extended to all hypotheses. This constitutes then
the numerical part of evidence theory. Finally, combination of evidence based
on the combination of bodies of arguments is discussed and a generalized
version of Dempster's rule is derived. The approach to evidence theory proposed
is general and is not limited to finite frames.

frameworks - DSmT

**+ Foundations
for a new theory of plausible and paradoxical reasoning**

J Dezert - Information and Security, (Tzv. Semerdjiev Editor) Bulgarian
Academy of Sciences, Sofia, 2002

This paper brings
foundations for a new theory of plausible and paradoxical reasoning and
describes a rule of combination of sources of information in a very general
framework where information can be both uncertain and paradoxical. In this new
theory, the rule of combination which takes into account explicitly both
conjunctions and disjunctions of assertions in the fusion process, appears to
be more simple and general than the Dempster’s rule of combination. Through
several simple examples, we show the strong ability of this new theory to solve
practical but difficult problems where the Dempster-Shafer theory usually
fails.

Reliability - logic

+ **A generalization of the algorithm of Heidtmann to non-monotone formulas**

R Bertschy, PA Monney - Journal of
Computational and Applied Mathematics, Volume 76, Issues 1–2, 17 December 1996,
Pages 55–76

http://www.sciencedirect.com/science/article/pii/S0377042796000891

The following
problem from reliability theory is considered. Given a disjunctive normal form
(DNF) *ϕ* = *ϕ*_{1}∨ … ∨*ϕr*, we
want to find a representation of ϕ into *disjoint* formulas,
i.e. find formulas *i* ≠ *j*. In addition, the
formulas ** **

fusion – classification – machine learning

+ **A comparison of two approaches for combining the votes of cooperating
classifiers**

J Franke, E Mandler - 11th IAPR International Conference on Pattern
Recognition, 1992. Vol. II, pages 611-614

Two different approaches are described to combine the results
of different classifiers. The first approach is based on the Dempster/Shafer
theory of evidence and the second one is a statistical approach with some
assumptions on the input data. Both approaches were tested for user-dependent
recognition of on-line handwritten characters.

Cited by __90__

information

**Requirements for total uncertainty measures in Dempster–Shafer theory of evidence**

J Abellán, A Masegosa - International Journal of General Systems 37, Issue 6, pages
733-747, 2008

http://www.tandfonline.com/doi/abs/10.1080/03081070802082486?journalCode=ggen20#.VRLWpvmsV7g

Recently, an alternative measure of total uncertainty
in Dempster–Shafer theory of evidence (DST) has been proposed in place of the
maximum entropy measure. It is based on the pignistic probability of a basic
probability assignment and it is proved that this measure verifies a set of
needed properties for such a type of measure. The proposed measure is motivated
by the problems that maximum (upper) entropy has. In this paper, we analyse the
requirements, presented in the literature, for total uncertainty measures in
DST and the shortcomings found on them. We extend the set of requirements,
which we consider as a set of requirements of properties, and we use the set of
shortcomings found on them to define a set of requirements of the behaviour for
total uncertainty measures in DST. We present the differences of the principal
total uncertainty measures presented in DST taking into account their
properties and behaviour.

applications – fusion - image

+ **Color image segmentation using the Dempster-Shafer theory of evidence for the fusion
of texture**

JB Mena, JA Malpica - ISPRS Archives, Vol. XXXIV, Part 3/W8, Munich,
17.-19. Sept. 2003

We present a new method for the
segmentation of color images for extracting information from terrestrial,
aerial or satellite images. It is a supervised method for solving a part of the
automatic extraction problem. The basic technique consists in fusing
information coming from three different sources for the same image. The first
source uses the information stored in each pixel, by means of the Mahalanobis
distance. The second uses the multidimensional distribution of the three bands
in a window centred in each pixel, using the Bhattacharyya distance. The last
source also uses the Bhattacharyya distance, in this case coocurrence matrices
are compared over the cube texture built around each pixel. Each source
represent a different order of statistic. The Dempster - Shafer theory of
evidence is applied in order to fuse the information from these three sources.
This method shows the importance of applying context

and textural properties for the
extraction process. The results prove the potential of the method for real
images starting from the three RGB bands only. Finally, some examples about the
extraction of linear cartographic features, specially roads, are shown.

continuous

+ **Belief functions on real numbers**

P Smets - International Journal of Approximate Reasoning 40, Issue 3,
November 2005, Pages 181–223

http://www.sciencedirect.com/science/article/pii/S0888613X0500023X

We generalize the TBM (transferable belief model)
to the case where the frame of discernment is the extended set of real
numbers R=[-∞,∞], under the assumptions that ‘masses’ can only
be given to intervals. Masses become densities, belief functions, plausibility
functions and commonality functions become integrals of these densities and
pignistic probabilities become pignistic densities. The mathematics of belief
functions become essentially the mathematics of probability density functions
on R^{2}.

applications - uncertainty - measurement

+ **Fully comprehensive mathematical approach to the expression of
uncertainty in measurement**

A Ferrero, S Salicone - IEEE Transactions on Instrumentation and Measurement, Volume 55,
Issue 3, pages 706 – 712, June 2006

This paper analyzes the result of a measurement in
the mathematical model of incomplete knowledge and shows how it can be treated
in the framework of the theory of evidence. The random fuzzy variables are
considered in order to express the result of a measurement together with its
uncertainty, and a significant application is considered to prove the practical
utility of the proposed approach.

debate

+ **The context model: An integrating view of vagueness and uncertainty**

J Gebhardt, R Kruse - International Journal of Approximate Reasoning 9, Issue 3,
October 1993, Pages 283–314

http://www.sciencedirect.com/science/article/pii/0888613X93900145

The problem
of handling vagueness and uncertainty as two different kinds of partial
ignorance has become a major issue in several fields of scientific research.
Currently the most popular approaches are Bayes theory, Shafer's evidence
theory, the transferable belief model, and the possibility theory with its
relationships to fuzzy sets.

Since the
justification of some of the mentioned theories has not been clarified for a
long time, some criticism on these models is still pending. For that reason we
have developed a model of vagueness and uncertainty—called the context
model—that provides a formal environment for the comparison and semantic
foundation of the referenced theories.

In this paper
we restrict ourselves to the presentation of basic ideas keyed to the
interpretation of Bayes theory and the Dempster-Shafer theory within the
context model. Furthermore the context model is applied to show a direct
comparison of these two approaches based on the well-known spoiled sandwich
effect, the three prisoners problem, and the unreliable alarm paradigm.

debate – survey – expected utility - uncertainty

+ **Recent
developments in modeling preferences: Uncertainty and ambiguity**

C Camerer, M Weber - Journal of Risk and Uncertainty, Volume 5, Issue 4,
pp 325-370, 1992

http://link.springer.com/article/10.1007%2FBF00122575

In subjective expected utility (SEU), the decision
weights people attach to events are their beliefs about the likelihood of
events. Much empirical evidence, inspired by Ellsberg (1961) and others, shows
that people prefer to bet on events they know more about, even when their
beliefs are held constant. (They are averse to *ambiguity*, or
uncertainty about probability.) We review evidence, recent theoretical
explanations, and applications of research on ambiguity and SEU.

decision making - utility

+ **On ϱ in a decision-theoretic apparatus of Dempster-Shafer theory**

J Schubert - International Journal of Approximate Reasoning 13, Issue 3,
October 1995, Pages 185–200

Thomas M. Strat has developed a decision-theoretic
apparatus for Dempster-Shafer theory (Decision analysis using belief
functions, *Intern. J. Approx. Reason.* 4(5/6), 391–417, 1990).
In this apparatus, expected utility intervals are constructed for different
choices. The choice with the highest expected utility is preferable to others.
However, to find the preferred choice when the expected utility interval of one
choice is included in that of another, it is necessary to interpolate a
discerning point in the intervals. This is done by the parameter ϱ,
defined as the probability that the ambiguity about the utility of every
nonsingleton focal element will turn out as favorable as possible. If there are
several different decision makers, we might sometimes be more interested in
having the highest expected utility among the decision makers rather than only
trying to maximize our own expected utility regardless of choices made by other
decision makers. The preference of each choice is then determined by the
probability of yielding the highest expected utility. This probability is equal
to the maximal interval length of ϱ under which an alternative is
preferred. We must here take into account not only the choices already made by
other decision makers but also the rational choices we can assume to be made by
later decision makers. In Strats apparatus, an assumption, unwarranted by the
evidence at hand, has to be made about the value of ϱ. We demonstrate
that no such assumption is necessary. It is sufficient to assume a uniform
probability distribution for ϱ to be able to discern the most preferable
choice. We discuss when this approach is justifiable.

transformation - approximation

+ **Approximation techniques for the transformation of fuzzy sets into
random sets**

MC Florea, AL Jousselme, D Grenier, É Bossé - Fuzzy Sets and Systems 159,
Issue 3, 1 February 2008, Pages 270–288

http://www.sciencedirect.com/science/article/pii/S0165011407004174

With the recent rising of numerous theories for
dealing with uncertain pieces of information, the problem of connection between
different frames has become an issue. In particular, questions such as how to
combine fuzzy sets with belief functions or probability measures often emerge.
The alternative is either to define transformations between theories, or to use
a general or unified framework in which all these theories can be framed.
Random set theory has been proposed as such a unified framework in which at
least probability theory, evidence theory, possibility theory and fuzzy set
theory can be represented. Whereas the transformations of belief functions or
probability distributions into random sets are trivial, the transformations of
fuzzy sets or possibility distributions into random sets lead to some issues.
This paper is concerned with the transformation of fuzzy membership functions
into random sets. In practice, this transformation involves the creation of a
large number of focal elements (subsets with non-null probability) based on the
α-cuts of the fuzzy membership functions. In order to keep a
computationally tractable fusion process, the large number of focal elements
needs to be reduced by approximation techniques. In this paper, we propose
three approximation techniques and compare them to classical approximations
techniques used in evidence theory. The quality of the approximations is
quantified using a distance between two random sets.

approximation - computation

+ **Resource bounded and anytime approximation of belief function
computations**

R Haenni, N Lehmann - International Journal of Approximate Reasoning 31,
Issues 1–2, October 2002, Pages 103–154

http://www.sciencedirect.com/science/article/pii/S0888613X02000749

This paper proposes a new approximation method for
Dempster–Shafer belief functions. The method is based on a new concept of
incomplete belief potentials. It allows to compute simultaneously lower and
upper bounds for belief and plausibility. Furthermore, it can be used for a
resource-bounded propagation scheme, in which the user determines in advance
the maximal time available for the computation. This leads then to convenient,
interruptible anytime algorithms giving progressively better solutions as
execution time goes on, thus offering to trade the quality of results against
the costs of computation. The paper demonstrates the usefulness of these new
methods and shows its advantages and drawbacks compared to existing techniques.

independence

+ **Independence concepts in evidence theory**

I Couso, S Moral - International Journal of Approximate Reasoning 51, Issue 7,
September 2010, Pages 748–758

http://www.sciencedirect.com/science/article/pii/S0888613X10000447

We study three conditions of independence within
evidence theory framework. The first condition refers to the selection of pairs
of focal sets. The remaining two ones are related to the choice of a pair of
elements, once a pair of focal sets has been selected. These three concepts
allow us to formalize the ideas of lack of interaction among variables and
among their (imprecise) observations. We illustrate the difference between both
types of independence with simple examples about drawing balls from urns. We
show that there are no implication relationships between both of them. We also
study the relationships between the concepts of “independence in the selection”
and “random set independence”, showing that they cannot be simultaneously
satisfied, except in some very particular cases.

debate – uncertainty

+ **On quantification of different facets of uncertainty**

NR Pal - Fuzzy Sets and Systems 107, Issue 1, 1 October 1999, Pages
81–91

http://www.sciencedirect.com/science/article/pii/S0165011498000050

With a brief introduction to three major types of
uncertainties, randomness, nonspecificity and fuzziness we discuss various
attempts to quantify them. We also discuss several attempts to quantify total
uncertainty in a system. We then talk about some new facets of uncertainty
like, higher-order fuzzy entropy, hybrid entropy and conflict in a body of
evidence. In conclusion, we indicate some other aspects of uncertainty that
need to be modeled and quantified.

Graphical models - markov

+ **Multisensor triplet Markov fields and theory of evidence**

W Pieczynski, D Benboudjema - Image and Vision Computing, Volume 24,
Issue 1, 1 January 2006, Pages 61–69

http://www.sciencedirect.com/science/article/pii/S0262885605001678

Hidden Markov Fields (HMF) are widely applicable to
various problems of image processing. In such models, the hidden process of
interest *X* is a Markov field, which must be estimated from its
observable noisy version *Y*. The success of HMF is due mainly to the
fact that *X* remains Markov conditionally on the observed
process, which facilitates different processing strategies such as Bayesian
segmentation. Such models have been recently generalized to ‘Pairwise’ Markov
fields (PMF), which offer similar processing advantages and superior modeling
capabilities. In this generalization, one directly assumes the Markovianity of
the pair (*X*,*Y*). Afterwards, ‘Triplet’ Markov fields (TMF) have
been proposed, in which the distribution of (*X*,*Y*) is the marginal
distribution of a Markov field (*X*,*U*,*Y*), where *U* is
an auxiliary random field. So *U* can have different
interpretations and, when the set of its values is not too complex, *X* can
still be estimated from *Y*. The aim of this paper is to show some
connections between TMF and the Dempster–Shafer theory of evidence. It is shown
that TMF allow one to perform the Dempster–Shafer fusion in different general
situations, possibly involving several sensors. As a consequence, Bayesian
segmentation strategies remain applicable.

Machine learning – decision trees

+ **Induction of decision trees from partially classified data using belief
functions**

T Denoeux, MS Bjanger - Master's thesis, Norwegian Univ. of Science and
Technology, Dpt of Computer and Information Science, 2000

A
new tree-structured classifier based on the Dempster-Shafer theory of evidence
is presented. The entropy measure classically used to assess the impurity of
nodes in decision trees is replaced by an evidence-theoretic uncertainty
measure taking into account not only the class proportions, but also the number
of objects in each node. The resulting algorithm allows the processing of
training data whose class membership is only partially specified in the form of
a belief function. Experimental results with EEG data are presented.

In
this paper, the problem of learning from partially classified data is addressed
from a different perspective using a new approach to decision tree (DT)
induction based on the theory of belief functions [1].

debate - TBM

**+ What
is Dempster-Shafer's model?**

P Smets - RR Yager, M. Fedrizzi, J.
Kacprzyk (Eds.), Advances in The Dempster-Shafer Theory of Evidence, Wiley, San
Mateo, CA (1994), pp. 5–34

Several
mathematical models have been proposed for the modelling of someone's degrees
of belief. The oldest is the Bayesian model that uses probability functions.
The upper and lower probabilities (ULP) model, Dempster's model, the
evidentiary value model (EVM) and the probability of modal propositions somehow
generalize the Bayesian approach. The transferable belief model (TBM) is based
on other premises and uses belief functions. None of these models is THE best: each
has its own domain of application. We spell out through examples what are the
underlying hypotheses that lead to the selection of an adequate model for a
given problem. We give indications on how to choose the appropriate model. The
major discriminating criterion is: if there exists a probability measure with
known values, use the Bayesian model, if there exists a probability measure but
with some unknown values, use the ULP models, if the existence of a probability
measure is not known, use the TBM. Dempster's model is essentially a special
case of ULP model. The EVM and the probability of modal propositions
(provability, necessity...) corresponds to a special use of the Bayesian model.

Cited by __197____ __

applications - fusion - reliability

O Basir, X Yuan - Information Fusion 8, Issue 4, October 2007, Pages 379–386

http://www.sciencedirect.com/science/article/pii/S156625350500076X

obasir@uwaterloo.ca

Engine
diagnostics is a typical multi-sensor fusion problem. It involves the use of
multi-sensor information such as vibration, sound, pressure and temperature, to
detect and identify engine faults. From the viewpoint of evidence theory, information
obtained from each sensor can be considered as a piece of evidence, and as
such, multi-sensor based engine diagnosis can be viewed as a problem of
evidence fusion. In this paper we investigate the use of Dempster–Shafer evidence theory as a tool for modeling and fusing
multi-sensory pieces of evidence pertinent to engine quality. We present a
preliminary review of Evidence Theory and explain how the multi-sensor engine
diagnosis problem can be framed in the context of this theory, in terms of faults
frame of discernment, mass functions and the rule for combining pieces of
evidence. We introduce two new methods for enhancing the effectiveness of mass
functions in modeling and combining pieces of evidence. Furthermore, we propose
a rule for making rational decisions with respect to engine quality, and
present a criterion to evaluate the performance of the proposed information
fusion system. Finally, we report a case study to demonstrate the efficacy of
this system in dealing with imprecise information cues and conflicts that may
arise among the sensors.

combination - conflict

- **Combining belief functions when evidence conflicts**

CK Murphy - Decision support systems, Volume 29, Issue 1, July 2000,
Pages 1–9

http://www.sciencedirect.com/science/article/pii/S0167923699000846

The
use of belief functions to represent and to manipulate uncertainty in expert
systems has been advocated by some practitioners and researchers. Others have
provided examples of counter-intuitive results produced by Dempster's rule for
combining belief functions and have proposed several alternatives to this rule.
This paper presents another problem, the failure to balance multiple evidence,
then illustrates the proposed solutions and describes their limitations. Of the
proposed methods, averaging best solves the normalization problems, but it does
not offer convergence toward certainty, nor a probabilistic basis. To achieve
convergence, this research suggests incorporating average belief into the
combining rule.

machine learning – classification - combination

+ **The combination of multiple classifiers using an evidential reasoning
approach**

Y Bi, J Guan, D Bell - Artificial Intelligence 172, Issue 15, October
2008, Pages 1731–1751

http://www.sciencedirect.com/science/article/pii/S0004370208000799

j.guan@qub.ac.uk, da.bell@qub.ac.uk

In many domains when we have several competing
classifiers available we want to synthesize them or some of them to get a more
accurate classifier by a combination function. In this paper we propose a
‘class-indifferent’ method for combining classifier decisions represented by
evidential structures called triplet and quartet, using Dempster's rule of
combination. This method is unique in that it distinguishes important elements
from the trivial ones in representing classifier decisions, makes use of more
information than others in calculating the support for class labels and
provides a practical way to apply the theoretically appealing Dempster–Shafer
theory of evidence to the problem of ensemble learning. We present a formalism
for modelling classifier decisions as triplet mass functions and we establish a
range of formulae for combining these mass functions in order to arrive at a
consensus decision. In addition we carry out a comparative study with the
alternatives of simplet and dichotomous structure and also compare two combination
methods, Dempster's rule and majority voting, over the UCI benchmark data, to
demonstrate the advantage our approach offers.

Cited by __52____ __

combination

**Quasi-associative operations in the combination of evidence**

RR Yager - Kybernetes, Vol. 16, Issue 1, pp. 37–41, 1987

http://www.emeraldinsight.com/doi/abs/10.1108/eb005755

Quasi‐associative operators are defined and
suggested as a general structure useful for representing a class of operators
used to combine various pieces of evidence. We show that both averaging
operators and Dempster‐Shafer combining operators can be represented in
this new framework.

combination

**A modified combination rule of evidence theory**

Y Deng, W-K Shi - Journal-Shanghai Jiaotong University, 2003

http://en.cnki.com.cn/Article_en/CJFDTOTAL-SHJT200308031.htm

Evidence theory is widely used in data fusion
systems. However, there exist some problems in its combination rule. A modified
combination rule was presented. First, a global conflict is calculated by the
weighted average of the local conflict. Then, a validity coefficient is defined
to show the effect of the conflict evidence on the combination results.
Compared with other combination rules, the proposed rule considers not only the
consistency of evidence, but also the conflict of them. The numerical example
shows that the new combination rule improves the reliability and rationality of
the combination results.

debate - entropy

+ **The principle of minimum specificity as a basis for evidential reasoning**

D Dubois, H Prade - Uncertainty in knowledge-based systems, Lecture Notes in
Computer Science Volume 286, 1987, pp 75-84

The framework of evidence theory is used to
represent uncertainty pervading a set of statements which refer to subsets of a
universe. Grades of credibility and plausibility attached to statements specify
a class of bodies of evidence. Using newly appeared measures of specificity, a
principle is stated in order to select, among these bodies of evidence, the one
which suitably represents the available information in the least arbitrary way.
It is shown that this principle, which is similar to the maximum entropy
principle, leads to a deductive reasoning approach under uncertainty, and also provides
a rule of combination which does not presuppose any independence assumption.
Particularly, it is more general than Dempster's.

combination - frameworks

**Generalized union and project operations for pooling uncertain and
imprecise information**

DA Bell, JW Guan, SK Lee - Data & Knowledge Engineering, Volume 18, Issue 2, March
1996, Pages 89–117

http://www.sciencedirect.com/science/article/pii/0169023X9500029R

We have
previously proposed an extended relational data model with the objective of
supporting uncertain information in a consistent and coherent manner. The
model, which can represent both uncertainty and imprecision in data, is based
on the Dempster-Shafer (D-S) theory of evidence, and it uses *bel* and *pls
*functions of the theory, with their definitions extended somewhat for this
purpose. Relational operations such as Select, Cartesian Product, Join,
Project, Intersect, and Union have previously been defined [21].

In this paper
we consider two data combination problems associated with the data model. These
problems are believed to be inherent in most database models which handle
uncertain information. The problems are: the potential existence in the
database of identical tuples which have different respective degrees of belief
(the *redundancy* problem), and the potential existence of
different tuples with the same key values (the *inconsistency* problem).
The redundancy problem was treated to some extent in an earlier paper, but the
inconsistency problem has not been considered at all yet.

Now the
well-known *orthogonal sum operation* in the D-S theory, which
performs the pooling of data for the purpose of making choices between
hypotheses, may be viewed as a means of reducing inconsistency in data arising
from different sources. This capability has not yet been exploited in our data
model. So the idea here is to define a new *combine* operation
as a primitive for handling inconsistency in relations.

When data
from a number of sources is being pooled — often in order to support decision
making — the Union operation, and the Project operation, are very important. We
are particularly interested in the case where tuples in operand relations match
attribute-wise, but have different uncertainty and imprecision characteristics.
The execution of both the Union and Project operations, which the new *combine* operation
can help solve, is a means of dealing with the problem of information
aggregation. We use the orthogonal sum, which generalizes results from
traditional probability theory in a natural and correct manner, for pooling
evidence during the *combine* computation.

The paper
also addresses the execution efficiency of our suggested approach. The
orthogonal sum operation is exponentially complex if implemented naively. A
linear time algorithm can readily be made available for Union and Project for
the simple case where the attribute values to be combined are singletons (i.e.,
atomic values — as in the conventional relational model). However, many
potential applications of the approach can exploit the new data model's
facility of supporting set-valued attributes. In the method presented here we
can combine data supporting non-singleton subsets in linear-time.

applications - medical

MJ Beynon, L Jones, CA Holt - IEEE Transactions on Systems, Man, And Cybernetics—Part A:
Systems and Humans, Vol. 36, No. 1, pages 173-186, January 2006

In this paper, a novel object classification method
is introduced and developed within a biomechanical study of human knee function
in which subjects are classified to one of two groups: subjects with
osteoarthritic (OA) and normal (NL) knee function. Knee-function
characteristics are collected using a three-dimensional motion-analysis
technique. The classification method transforms these characteristics into sets
of three belief values: a level of belief that a subject has OA knee function,
a level of belief that a subject has NL knee function, and an associated level
of uncertainty. The evidence from each characteristic is then combined into a
final set of belief values, which is used to classify subjects. The final
belief values are subsequently represented on a simplex plot, which enables the
classification of a subject to be represented visually. The control parameters,
which are intrinsic to the classification method, can be chosen by an expert or
by an optimization approach. Using a leave-one-out cross-validation approach,
the classification accuracy of the proposed method is shown to compare
favorably with that of a well-established classifier-linear discriminant
analysis. Overall, this study introduces a visual tool that can be used to
support orthopaedic surgeons when making clinical decisions.

debate – foundations - statistics

**A theory of statistical evidence**

G Shafer - Foundations of Probability Theory, Statistical Inference, and
Statistical Theories of Science, The University of Western Ontario Series in
Philosophy of Science Volume 6b, 1976, pp 365-436

http://link.springer.com/chapter/10.1007%2F978-94-010-1436-6_11

There are at least two ways in which the impact of
evidence on a proposition may vary. On the one hand, there are various possible
degrees to which the evidence may support the proposition: taken as a whole, it
may support it strongly, just a little, or not at all. On the other hand, there
are various possible degrees to which the evidence may cast doubt on the
proposition: taken as a whole, it may cast serious doubt on it, thus rendering
it extremely doubtful or implausible; it may cast only moderate doubt on it,
thus leaving it moderately plausible; or it may cast hardly any doubt on it,
thus leaving it entirely plausible.

other theories – intuitionistic fuzzy sets - decision

+ **An interpretation of intuitionistic fuzzy sets in terms of evidence
theory: Decision making aspect**

L Dymova, P Sevastjanov - Knowledge-Based Systems, Volume 23, Issue 8,
December 2010, Pages 772–782

This paper presents a new interpretation of
intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of
evidence (*DST*). This interpretation makes it possible to represent all
mathematical operations on intuitionistic fuzzy values as the operations on
belief intervals. Such approach allows us to use directly the Dempster’s rule
of combination to aggregate local criteria presented by intuitionistic fuzzy
values in the decision making problem. The usefulness of the developed method
is illustrated with the known example of multiple criteria decision making
problem. The proposed approach and a new method for interval comparison based
on *DST*, allow us to solve multiple criteria decision making problem
without intermediate defuzzification when not only criteria, but their weights
are intuitionistic fuzzy values.

Cited by __62____ __

combination

+ **On the aggregation of prioritized belief structures**

RR Yager - IEEE Transactions on Systems, Man, and Cybernetics - Part A:
Systems and Humans, Vol 26, No. 6, Pages 708-717, November 1996

The focus of this work is to provide a procedure
for aggregating prioritized belief structures. Motivated by the ideas of
nonmonotonic logics an alternative to the normalization step used in Dempster's
rule when faced with conflicting belief structures is suggested. We show how
this procedure allows us to make inferences in inheritance networks where the
knowledge is in the form of a belief structure.

machine learning - regression

+ **Nonparametric regression analysis of uncertain and imprecise data using
belief functions**

S Petit-Renaud, T Denśux - International Journal of Approximate Reasoning
35, Issue 1, January 2004, Pages 1–28

http://www.sciencedirect.com/science/article/pii/S0888613X03000562

This paper introduces a new approach to regression
analysis based on a fuzzy extension of belief function theory. For a given
input vector *y*, in the form of a fuzzy belief assignment (FBA),
defined as a collection of fuzzy sets of values with associated masses of
belief. The output FBA is computed using a nonparametric, instance-based
approach: training samples in the neighborhood of

Cited by __81____ __

other theories – survey - previsions

+ **A survey of the theory of coherent lower previsions**

E Miranda - International Journal of Approximate Reasoning 48, Issue 2,
June 2008, Pages 628–658

http://www.sciencedirect.com/science/article/pii/S0888613X07001867

This paper presents a summary of Peter Walley’s
theory of coherent lower previsions. We introduce three representations of
coherent assessments: coherent lower and upper previsions, closed and convex
sets of linear previsions, and sets of desirable gambles. We show also how the
notion of coherence can be used to update our beliefs with new information, and
a number of possibilities to model the notion of independence with coherent
lower previsions. Next, we comment on the connection with other approaches in
the literature: de Finetti’s and Williams’ earlier work, Kuznetsov’s and
Weischelberger’s work on interval-valued probabilities, Dempster–Shafer theory
of evidence and Shafer and Vovk’s game-theoretic approach. Finally, we present
a brief survey of some applications and summarize the main strengths and
challenges of the theory.

Cited by __76__

information retrieval - applications

**Situation theory and Dempster-Shafer's theory of evidence for information retrieval**

M Lalmas, CJ Van Rijsbergen - Incompleteness and Uncertainty in Information Systems, Workshops
in Computing 1994, pp 102-116

http://link.springer.com/chapter/10.1007%2F978-1-4471-3242-4_8

We propose a model of information retrieval systems
that is based on a Theory of Information and a Theory of Uncertainty,
respectively Situation Theory and Dempster-Shafer’s Theory of Evidence. These
were selected because they allow us to tackle two of the main problems that
confront any attempt to model an information retrieval system: the
representation of information and its flow; and the uncertainty engendered by
the complexity and ambiguity arising when dealing with information.

debate – combination - evidence

G Shafer - International Journal of Intelligent Systems 1, Issue 3,
pages 155–179, Autumn (Fall) 1986

http://onlinelibrary.wiley.com/doi/10.1002/int.4550010302/abstract

This article provides a historical and conceptual
perspective on the contrast between the Bayesian and belief function approaches
to the probabilistic combination of evidence. It emphasizes the simplest
example of non-Bayesian belief-function combination of evidence, which was
developed by Hooper in the 1680s.

applications – information systems - risk

L Sun, RP Srivastava, TJ Mock - Journal of Management Information Systems, Vol 22, No 4,
pages 109-142, 2006

http://www.tandfonline.com/doi/abs/10.2753/MIS0742-1222220405#.VRL2OfmsV7g

This study develops an alternative methodology for
the risk analysis of information systems security (ISS), an evidential
reasoning approach under the Dempster-Shafer theory of belief functions. The
approach has the following important dimensions. First, the evidential
reasoning approach provides a rigorous, structured manner to incorporate
relevant ISS risk factors, related countermeasures, and their
interrelationships when estimating ISS risk. Second, the methodology employs
the belief function definition of risk--that is, ISS risk is the plausibility
of ISS failures. The proposed approach has other appealing features, such as
facilitating cost- benefit analyses to help promote efficient ISS risk
management. The paper elaborates the theoretical concepts and provides
operational guidance for implementing the method. The method is illustrated
using a hypothetical example from the perspective of management and a
real-world example from the perspective of external assurance providers.
Sensitivity analyses are performed to evaluate the impact of important
parameters on the model's results.

applications – robotics - navigation

+ **An improved map-matching algorithm used in vehicle navigation system**

D Yang, B Cai, Y Yuan – Proceedings of Intelligent Transportation
Systems, Vol 2, pages 1246-1250, 2003 DOI:10.1109/ITSC.2003.1252683

Vehicle navigation system estimates vehicle
location from Global Positioning System (GPS) or dead-reckoning (DR) system.
However, because of unknown GPS noise, the estimated location has undesirable
errors. The purpose of map matching (MM) algorithm is to locate the inaccuracy
position of the vehicle to the map data that is referenced by the system. In
this paper, a simple map-matching algorithm is discussed first. Then Kalman
filtering and Dempster-Shafer (D-S) theory of evidence are introduced into the
improved map-matching algorithm. The real road experiments demonstrate the effectiveness
and applicability of the improved algorithm, and it is found to produce better
results.

applications – semantic web

+ **Using the Dempster-Shafer theory
of evidence to resolve ABox inconsistencies**

A Nikolov, V Uren, E Motta, A De Roeck - Uncertainty Reasoning for the Semantic Web I, Lecture
Notes in Computer Science Volume 5327, 2008, pp 143-160

Automated ontology population using information
extraction algorithms can produce inconsistent knowledge bases. Confidence values
assigned by the extraction algorithms may serve as evidence in helping to
repair inconsistencies. The Dempster-Shafer theory of evidence is a formalism,
which allows appropriate interpretation of extractors’ confidence values. This
chapter presents an algorithm for translating the subontologies containing
conflicts into belief propagation networks and repairing conflicts based on the
Dempster-Shafer plausibility.

Cited by __20__

other theories

+ **A general non-probabilistic
theory of inductive reasoning**

W Spohn - arXiv preprint arXiv:1304.2375, 2013 - arxiv.org

Probability theory, epistemically interpreted, provides an
excellent, if not the best available account of inductive reasoning. This is so
because there are general and definite rules for the change of subjective probabilities
through information or experience; induction and belief change are one and same
topic, after all. The most basic of these rules is simply to conditionalize
with respect to the information received; and there are similar and more
general rules. 1 Hence, a fundamental reason for the epistemological success of
probability theory is that there at all exists a well-behaved concept of
conditional probability. Still, people have, and have reasons for, various
concerns over probability theory. One of these is my starting point:
Intuitively, we have the notion of plain belief; we believe propositions2 to be
true (or to be false or neither). Probability theory, however, offers no formal
counterpart to this notion. Believing A is not the same as having probability 1
for A, because probability 1 is incorrigible3; but plain belief is clearly
corrigible. And believing A is not the same as giving A a probability larger
than some 1 - c, because believing A and believing B is usually taken to be
equivalent to believing A & B.4 Thus, it seems that the formal
representation of plain belief has to take a non-probabilistic route. Indeed,
representing plain belief seems easy enough: simply represent an epistemic
state by the set of all propositions believed true in it or, since I make the
common assumption that plain belief is deductively closed, by the conjunction
of all propositions believed true in it. But this does not yet provide a theory
of induction, i.e. an answer to the question how epistemic states so represented
are changed tbrough information or experience. There is a convincing partial
answer: if the new information is compatible with the old epistemic state, then
the new epistemic state is simply represented by the conjunction of the new
information and the old beliefs. This answer is partial because it does not
cover the quite common case where the new information is incompatible with the
old beliefs. It is, however, important to complete the answer and to cover this
case, too; otherwise, we would not represent plain belief as conigible. The
crucial problem is that there is no good completion. When epistemic states are
represented simply by the conjunction of all propositions believed true in it,
the answer cannot be completed; and though there is a lot of fruitful work, no
other representation of epistemic states has been proposed, as far as I know,
which provides a complete solution to this problem. In this paper, I want to
suggest such a solution. In [4], I have more fully argued that this is the only
solution, if certain plausible desiderata are to be satisfied. Here, in section
2, I will be content with formally defining and intuitively explaining my
proposal. I will compare my proposal with probability theory in section 3. It
will turn out that the theory I am proposing is structurally homomorphic to
probability theory in important respects and that it is thus equally easily
implementable, but moreover computationally simpler. Section 4 contains a very
brief comparison with various kinds of logics, in particular conditional logic,
with Shackle's functions of potential surprise and related theories, and with
the Dempster - Shafer theory of belief functions.

applications - communications

+ **Intrusion detection engine based on Dempster-Shafer's theory of evidence**

W Hu, J Li, Q Gao – Proceedings of the 2006 International Conference on Communications,
Circuits and Systems, Vol 3, 1627-1631, 2006

In the decision making process, the uncertainty
existing in the network often leads to the failure of intrusion detection or
low detection rate. The Dempster-Shafer's theory of evidence in data fusion has
solved the problem of how to analyze the uncertainty in a quantitative way. In
the evaluation, the ingoing and outgoing traffic ratio and service rate are
selected as the detection metrics, and the prior knowledge in the DDoS domain
is proposed to assign probability to evidence. Furthermore, the combination
rule is used to combine the data collected by two sensors. The curves of belief
mass function varied with time are also shown in the paper. Finally, the
analysis of experimental results proves the ID detection engine efficient and
applicable. The conclusions provide us with the academic foundation for our
future implementation.

Cited by __20____ __

applications – retrieval – data mining

**EDM: a general framework for data mining based on evidence theory**

SS Anand, DA Bell, JG Hughes - Data & Knowledge Engineering, Volume 18,
Issue 3, April 1996, Pages 189–223

http://www.sciencedirect.com/science/article/pii/0169023X9500038T

Data Mining
or Knowledge Discovery in Databases [1, 15, 23] is currently one of the most
exciting and challenging areas where database techniques are coupled with
techniques from Artificial Intelligence and mathematical sub-disciplines to
great potential advantage. It has been defined as the non-trivial extraction of
implicit, previously unknown and potentially useful information from data. A
lot of research effort is being directed towards building tools for discovering
interesting patterns which are hidden below the surface in databases. However,
most of the work being done in this field has been problem-specific and no
general framework has yet been proposed for Data Mining. In this paper we seek
to remedy this by proposing, EDM — Evidence-based Data Mining — a general
framework for Data Mining based on Evidence Theory. Having a general framework
for Data Mining offers a number of advantages. It provides a common method for
representing knowledge which allows prior knowledge from the user or knowledge
discoveryd by another discovery process to be incorporated into the discovery
process. A common knowledge representation also supports the discovery of
meta-knowledge from knowledge discovered by different Data Mining techniques.
Furthermore, a general framework can provide facilities that are common to most
discovery processes, e.g. incorporating domain knowledge and dealing with
missing values.

The framework
presented in this paper has the following additional advantages. The framework
is inherently parallel. Thus, algorithms developed within this framework will
also be parallel and will therefore be expected to be efficient for large data
sets — a necessity as most commercial data sets, relational or otherwise, are
very large. This is compounded by the fact that the algorithms are complex.
Also, the parallelism within the framework allows its use in parallel,
distributed and heterogeneous databases. The framework is easily updated and
new discovery methods can be readily incorporated within the framework, making
it ‘general’ in the functional sense in addition to the representational sense
considered above. The framework provides an intuitive way of dealing with
missing data during the discovery process using the concept of Ignorance
borrowed from Evidence Theory.

The framework
consists of a method for representing data and knowledge, and methods for data
manipulation or knowledge discovery. We suggest an extension of the
conventional definition of mass functions in Evidence Theory for use in Data
Mining, as a means to represent evidence of the existence of rules in the
database. The discovery process within EDM consists of a series of operations
on the mass functions. Each operation is carried out by an EDM operator. We
provide a classification for the EDM operators based on the discovery functions
performed by them and discuss aspects of the induction, domain and combination
operator classes.

The
application of EDM to two separate Data Mining tasks is also addressed,
highlighting the advantages of using a general framework for Data Mining in
general and, in particular, using one that is based on Evidence Theory.

other theories – fuzzy - expert

- **Constructing fuzzy measures in expert systems**

GJ Klir, Z Wang, D Harmanec - Fuzzy sets and Systems, Volume 92, Issue
2, 1 December 1997, Pages 251–264

http://www.sciencedirect.com/science/article/pii/S0165011497001747

This paper is an overview of results regarding
various representations of fuzzy measures and methods for constructing fuzzy
measures in the context of expert systems, which were obtained by the authors
and their associates during the last three years. Included are methods for
constructing fuzzy measures by various transformations, by extension, by
statistical inference, and by various data-driven methods based either on the
Sugeno-integral or the Choquet-integral and using neural networks, genetic
algorithms, or fuzzy relation equations.

Cited by __100____ __

combination

**Hedging in the combination of evidence**

RR Yager - Journal of Information and Optimization Sciences 4, Issue 1,
pages 73-81, 1983

http://www.tandfonline.com/doi/abs/10.1080/02522667.1983.10698752#.VRL3P_msV7g

We discuss the dilemma suggested by Zadeh when using
Dempster's rule for combining evidence. We suggested a method for handling this
problem by the inclusion of a hedging element.

machine learning – decision trees

+ **Belief decision trees: theoretical foundations**

Z Elouedi, K Mellouli, P Smets - International Journal of Approximate Reasoning
28, No 2–3, November 2001, 91–124

http://www.sciencedirect.com/science/article/pii/S0888613X01000457

This paper extends the decision tree technique to
an uncertain environment where the uncertainty is represented by belief
functions as interpreted in the transferable belief model (TBM). This so-called
belief decision tree is a new classification method adapted to uncertain data.
We will be concerned with the construction of the belief decision tree from a
training set where the knowledge about the instances' classes is represented by
belief functions, and its use for the classification of new instances where the
knowledge about the attributes' values is represented by belief functions.

Cited by __93____ __

other theories - modal logic

- **Interpretations of various uncertainty theories using models of modal
logic: a summary**

G Resconi, GJ Klir, D Harmanec, U St Clair - Fuzzy Sets and Systems 80,
Issue 1, 27 May 1996, Pages 7–14

http://www.sciencedirect.com/science/article/pii/0165011495002626

This paper summarizes our efforts to establish the
usual semantics of propositional modal logic as a unifying framework for
various uncertainty theories. Interpretations for fuzzy set theory,
Dempster-Shafer theory, probability theory, and possibility theory are
discussed. Some properties of these interpretations are also presented, as well
as directions for future research.

fusion - logic

+ **Applying theory of
evidence in multisensor data fusion: a logical interpretation**

L Cholvy - Proceedings of FUSION 2000, Vol 1, pages TUB4/17 - TUB4/24,
2000 - DOI:10.1109/IFIC.2000.862670

Theory of evidence is a mathematical theory which
allows one to reason with uncertainty and which suggests a way for combining
uncertain data. This is the reason why it is used as a basic tool for
multisensor data fusion in the situation assessment process. Although numerical
people know this formalism quite well and its use in multisensor fusion, it is
not the case for people used to manipulating logical formalisms. The article
intends to give them the key for understanding Theory of Evidence and its use
in multisensor data fusion. It does this by giving a logical interpretation of
this formalism when the numbers are rational, and also by reformulating, in a
particular case, one model defined by A. Appriou for multisensor data fusion.

+ other theories – possibility - framework

**Evidence theory of exponential possibility distributions**

H Tanaka, H Ishibuchi - International Journal of Approximate Reasoning 8, Issue 2,
February 1993, 123–140

http://www.sciencedirect.com/science/article/pii/0888613X93900248

This paper studies a certain form of evidence
theory using exponential possibility distributions. Because possibility
distributions are obtained from an expert knowledge or can be identified from
given data, a possibility distribution is regarded as a representation of
evidence in this paper. A rule of combination of evidence is given similar to
Dempster's rule. Also, the measures of ignorance and fuzziness of evidence are
defined by a normality factor and the area of a possibility distribution,
respectively. These definitions are similar to those given by G. Shafer and A.
Kaufman et al., respectively. Next, marginal and conditional possibilities are
discussed from a joint possibility distribution, and it is shown that these
three definitions are well matched to each other. Thus, the posterior
possibility is derived from the prior possibility in the same form as Bayes'
formula. This fact shows the possibility that an information-decision theory
can be reconstructed from the viewpoint of possibility distributions.
Furthermore, linear systems whose variables are defined by possibility
distributions are discussed. Operations of fuzzy vectors defined by
multidimensional possibility distributions are well formulated, using the
extension principle of L. A. Zadeh.

applications – reliability - framework

X Fan, MJ Zuo - Pattern Recognition Letters, Volume 27, Issue 5, 1 April
2006, Pages 366–376

http://www.sciencedirect.com/science/article/pii/S0167865505002382

In this paper, conventional D–S evidence theory is improved through the
introduction of fuzzy membership function, importance index, and conflict
factor in order to address the issues of evidence sufficiency, evidence
importance, and conflicting evidences in the practical application of D–S
evidence theory. New decision rules based on the improved D–S evidence theory
are proposed. Examples are given to illustrate that the improved D–S evidence
theory is better able to perform fault diagnosis through fusing multi-source
information than conventional D–S evidence theory.

Cited by __90__

inference - uncertainty

**About assessing and evaluating uncertain inferences within the theory of evidence**

T Kämpke - Decision Support Systems, Volume 4, Issue 4, December 1988,
Pages 433–439

http://www.sciencedirect.com/science/article/pii/0167923688900061

Dealing with uncertainty of facts and rules in an
inference system will be discussed. The assessment and evaluation of
uncertainties will be done within Dempster's and Shafer's theory of evidence.
The relation between this theory and classical probability theory will be
stressed.

combination - expert

**+ [book] ****Aggregation and fusion of imperfect information**

B Bouchon (editor) – 1998

http://www.springer.com/gp/book/9783790810486

This book presents the main tools for aggregation
of information given by several members of a group or expressed in multiple
criteria, and for fusion of data provided by several sources. It focuses on the
case where the availability knowledge is imperfect, which means that uncertainty
and/or imprecision must be taken into account. The book contains both
theoretical and applied studies of aggregation and fusion methods in the main
frameworks: probability theory, evidence theory, fuzzy set and possibility
theory. The latter is more developed because it allows to manage both imprecise
and uncertain knowledge. Applications to decision-making, image processing,
control and classification are described. The reader can find a
state-of-the-art of the main methodologies, as well as more advanced results
and descriptions of utilizations.

applications - psychology

+ **An empirical evaluation of descriptive models of ambiguity reactions in
choice situations**

SP Curley, JF Yates - Journal of Mathematical Psychology, Volume 33,
Issue 4, December 1989, Pages 397–427

http://www.sciencedirect.com/science/article/pii/0022249689900199

Ambiguity is uncertainty about an option's
outcome-generating process, and is characterized as uncertainty about an
option's outcome probabilities. Subjects, in choice tasks, typically have
avoided ambiguous options. Descriptive models are identified and tested in two
studies which had subjects rank monetary lotteries according to preference. In
Study 1, lotteries involved receiving a positive amount or nothing, where *P* denotes
the probability of receiving the nonzero amount. Subjects were willing to
forego expected winnings to avoid ambiguity near *P* = .50
and *P* = .75. Near *P* = .25, a significant
percentage of subjects exhibited ambiguity seeking, with subjects, on average,
willing to forego expected winnings to have the more ambiguous option. The
observed behavior contradicts the viability of a proposed lexicographic model.
Study 2 tested four polynomial models using diagnostic properties in the
context of conjoint measurement theory. The results supported a sign dependence
of ambiguity with respect to the probability level *P*, such that
subjects' preference orderings over ambiguity reversed with changes in *P*.
This behavior was inconsistent with all the three-factor polynomial models
investigated. Further analyses failed to support a variant of portfolio theory,
as well. The implications of these results for the descriptive modeling of
choice under ambiguity are discussed.

applications – medical – classification - fusion

+ **Classifier fusion using Dempster-Shafer theory of evidence to predict breast cancer tumors**

M Raza, I Gondal, D Green and RL Coppel - Proc. 2006 IEEE Region 10 Conference (TENCON 2006)

DOI: 10.1109/TENCON.2006.343718

{Mansoor.Raza, Iqbal. Gondal, David. Green@,Infotech.monash.edu.au

Ross. Coppel@med. monash. edu. au

In classifier fusion models, classifiers outputs
are combined to achieve a group decision. The most often used classifiers
fusion models are majority vote, probability schemes, weighted averaging and
Bayes approach to name few. We propose a model of classifiers fusion by
combining the mathematical belief of classifiers. We used Dempster-Shafer
theory of evidence to determine the mathematical belief of classifiers. Support
vector machine (SVM) with linear, polynomial and radial kernel has been
employed as classifiers. The output of classifiers used as basis for computing
beliefs. We combined these beliefs to arrive at one final decision. Our experimental
results have shown that the new proposed classifiers fusion methodology have
outperforms single classification models.

combination - survey

**On the combination of evidence in various mathematical frameworks**

D Dubois, H Prade - Reliability Data Collection and Analysis, Eurocourses Volume 3,
1992, pp 213-241

http://link.springer.com/chapter/10.1007/978-94-011-2438-6_13

The problem of combining pieces of evidence issued
from several sources of information turns out to be a very important issue in
artificial intelligence. It is encountered in expert systems when several
production rules conclude on the value of the same variable, but also in robotics
when information coming from different sensors is to be aggregated. Solutions
proposed in the literature so far have often been unsatisfactory because
relying on a single theory of uncertainty, a unique mode of combination, or the
absence of analysis of the reasons for uncertainty. Besides dependencies and
redundancies between sources must be dealt with especially in knowledge bases,
where sources correspond to production rules.

geometry - conditioning

+ **Geometric analysis of belief space and conditional subspaces**

F Cuzzolin, R Frezza - Proceedings of the 2nd International Symposium on Imprecise
Probabilities and Their Applications (ISIPTA 2001), Ithaca, NY, 2001

In this paper the geometric structure of the space
S of the belief functions dened over a discrete set (belief space) is analyzed.
Using the Moebius inversion lemma we prove the recursive bundle structure of
the belief space and show how an arbitrary belief function can be uniquely
represented as a convex combination of certain elements of the bers, giving S
the form of a simplex. The commutativity of orthogonal sum and convex closure
operator is proved and used to depict the geometric structure of conditional
subspaces, i.e. sets of belief functions conditioned by a given function s.
Future applications of these geometric methods to classical problems like
probabilistic approximation and canonical decomposition are outlined.

other theories – rough sets

Z Pawlak, J Grzymala-Busse, R Slowinski and W Ziarko - Communications of the ACM, Volume 38 Issue 11, Nov.
1995, Pages 88-95

Rough set theory, introduced by Zdzislaw Pawlak in
the early 1980s [11, 12], is a new mathematical tool to deal with vagueness and
uncertainty. This approach seems to be of fundamental importance to artificial
intelligence (AI) and cognitive sciences, especially in the areas of machine
learning, knowledge acquisition, decision analysis, knowledge discovery from
databases, expert systems, decision support systems, inductive reasoning, and
pattern recognition.

The rough set concept overlaps—to some extent —with
many other mathematical tools developed to deal with vagueness and uncertainty,
in particular with the Dempster-Shafer theory of evidence [15].

The main difference is that the Dempster-Shafer
theory uses belief functions as a main tool, while rough set theory makes use
of sets—lower and upper approximations. Another relationship exists between
fuzzy set theory and rough set theory [13]. Rough set theory does not compete
with fuzzy set theory, with which it is frequently contrasted, but rather
complements it [1]. In any case, rough set theory and fuzzy set theory are
independent approaches to imperfect knowledge. Furthermore, some relationship
exists between rough set theory and discriminant analysis [7], Boolean
reasoning methods [16], and decision analysis [14].

One of the main advantages of rough set theory is
that it does not need any preliminary or additional information about data,
such as probability distribution in statistics, basic probability assignment in
the Dempster-Shafer theory, or grade of membership or the value of possibility
in fuzzy set theory [2].

Cited by 1__517__

conditioning – graphical models

+ **Binary join trees for computing marginals in the Shenoy-Shafer
architecture**

PP Shenoy - International Journal of Approximate Reasoning, Volume 17,
Issues 2–3, August–October 1997, pp 239–263

http://www.sciencedirect.com/science/article/pii/S0888613X97891359

We describe a data structure called binary join
trees that is useful in computing multiple marginals efficiently in the
Shenoy-Shafer architecture. We define binary join trees, describe their
utility, and describe a procedure for constructing them.

applications - risk

S Kaplan - Risk Analysis 17, Issue 4, pages 407–417, August 1997

http://onlinelibrary.wiley.com/doi/10.1111/j.1539-6924.1997.tb00881.x/abstract

… Also, there are a bunch of more recent theories
that have been invented to fix alleged deficiencies in the traditional ideas.
There’s Possibility Theory, Dempster/Shaefer Theory of Evidence, Higher Order
Probability Theory, etc. Notable among these, and currently in vogue, are the
fuzzy theories (e.g., Ref. 5), which attempt to encompass, in addition to the
traditional meanings, the notions of ambiguity, vagueness, lack of definition,
and also of paradoxes, such as the famous one about the barber who shaves those
and only those who do not shave themselves. Does this barber shave himself?
Well, if he does, he doesn’t, and if he doesn’t, he does. That’s the paradox.

Cited by __304 __

applications - image

**A method
for initial hypothesis formation in image understanding**

NB Lehrer, G Reynolds, J Griffith - Technical Report, University of
Massachusetts Amherst, MA, USA, 1987

http://adsabs.harvard.edu/abs/1980osa..meet...97L

This paper presents a method for initial hypothesis
formation in image understanding where the knowledge base is automatically
constructed given a set of training instances. The hypotheses formed by this
system are intended to provide an initial focus-of-attention set of objects
from a knowledge-directed, opportunistic image understanding system whose
intended goal is the interpretation of outdoor natural scenes. Presented is an
automated method for defining world knowledge based on the frequency distributions
of a set of training objects and feature measurements. This method takes into
consideration the imprecision (inaccurate feature measurements) and
incompleteness (possibly too few samples) of the statistical informa- tion
available from the training set. A computationally efficient approach to the
dempster-shafer theory of evidence is used for the representation and
combination of evidence from disparate sources. We chose the Dempster-Shafer
theory in order to take advantage of its rich representation of belief,
disbelief, uncertainty and conflict. A brief intuitive discussion of the Dempster-Shafer
theory of evidence is contained in appendix A.

debate - survey

+ **A
synthetic view of approximate reasoning techniques**

H Prade - Proceedings of the Eighth International Joint Conference on
Artificial intelligence (IJCAI'83), Volume 1, Pages 130-136, 1983

This paper presents a review of different
approximate reasoning techniques which have been proposed for dealing with
uncertain or imprecise knowledge, especially in expert systems based on
production rule methodology. Theoretical approaches such as Bayesian inference,
Shafer's belief theory or Zadeh's possibility theory as well as more empirical
proposals such as the ones used in MYCIN or in PROSPECTOR, are considered. The
presentation is focused on two basic inference schemes: the deductive inference
and the combination of several uncertain or imprecise evidences relative to a
same matter. Several kinds of uncertainty are taken into account in the models
which are described in the paper: different degrees of certainty or of truth
may be associated with the observed or produced facts or with the " if..,
then..." rules; moreover the statements of facts or of rules may be
imprecise or fuzzy and the values of the degrees of certainty which are used
may be only approximately known. An extensive bibliography, to which it is
referred in the text, is appended.

Cited by __61__

statistics

+ **The Dempster–Shafer calculus for statisticians**

AP Dempster - International Journal of Approximate Reasoning 48, Issue 2, June
2008, Pages 365–377

http://www.sciencedirect.com/science/article/pii/S0888613X07000278

The Dempster–Shafer (DS) theory of probabilistic
reasoning is presented in terms of a semantics whereby every meaningful formal
assertion is associated with a triple (p, q, r) where p is the probability
“for” the assertion, q is the probability “against” the assertion, and r is the
probability of “don’t know”. Arguments are presented for the necessity of
“don’t know”. Elements of the calculus are sketched, including the extension of
a DS model from a margin to a full state space, and DS combination of
independent DS uncertainty assessments on the full space. The methodology is
applied to inference and prediction from Poisson counts, including an
introduction to the use of join-tree model structure to simplify and shorten
computation. The relation of DS theory to statistical significance testing is
elaborated, introducing along the way the new concept of “dull” null
hypothesis.

applications - image

H Rasoulian, WE Thompson, LF Kazda and R Parra-Loera - Proc. SPIE 1310,
Signal and Image Processing Systems Performance Evaluation, 199 (September 1,
1990); doi:10.1117/12.21811

http://spie.org/Publications/Proceedings/Paper/10.1117/12.21811

In electronic vision systems, locating regions of
interest-a process referred to as cueing-allows the computing power of the
vision system to be focused on small regions rather than the entire scene. The
purpose of this paper is to illustrate the ability of a new technique to locate
regions that may contain objects of interest. This technique employs the
mathematical theory of evidence to combine evidence received from disparate
sources. Here the evidence consists of the images obtained from two sources: laser
radar range and laser radar amplitude. The mean values of the super pixel gray
levels for the two images are calculated and combined based on the
Dempster-Shafer rule of combination.

combination – other theories - fuzzy

- **Fuzzy set connectives as combinations of belief structures**

D Dubois, RR Yager - Information Sciences 66, Issue 3, 15 December 1992, Pages
245–276

http://www.sciencedirect.com/science/article/pii/002002559290096Q

Consonant belief structures provide a
representation for fuzzy sets owing to the fact that their plausibility
measures are essentially possibility measures. We note that two belief
structures are equivalent if their plausibility and belief functions are equal.
This observation leads us to provide a multiple number of equivalent
representations for any belief structure. Commensurate representations can be
induced for two different belief structures by forcing the same number of focal
elements with the same weights. We show that if we represent two consonant
belief structures in a commensurate manner then their aggregations are closed
with respect to consonance, provided that the additional requirement that the
underlying probability distributions satisfy a condition of correlation is
imposed. The results of this work allow us to use belief structure
representations for the manipulation of fuzzy subsets under various logical
combinations. All basic fuzzy set connectives can thus be interpreted in the
framework of the theory of evidence.

Cited by __30__

applications - geoscience

L Hubert-Moy, S Corgne, G Mercier and B Solaiman - Proceedings of the Fifth International Conference
on Information Fusion, Volume 1,
Page(s): 114 – 121, 2002

In intensive agricultural regions, accurate
assessment of the spatial and temporal variation of winter vegetation covering
is a key indicator of water transfer processes, essential for controlling land
management and helping local decision making. Spatial prediction modeling of
winter bare soils is complex and it is necessary to introduce uncertainty in modeling
land use and cover changes, especially as high spatial and temporal variability
are encountered. Dempster's fusion rule is used in the present study to
spatially predict the location of winter bare fields for the next season on a
watershed located in an intensive agricultural region. It expresses the model
as a function of past-observed bare soils, field size, distance from farm
buildings, agro-environmental action, and production quotas per ha. The model
well predicted the presence of bare soils on 4/5 of the total area. The spatial
distribution of misrepresented fields is a good indicator for identifying
change factors.

conditioning - entropy

+ **A method
of computing generalized Bayesian probability values for expert systems**

P Cheeseman - Proceedings of the Eighth International Joint Conference
on Artificial intelligence (IJCAI), Vol 1, 198-202

This paper presents a new method for calculating
the conditional probability of any multi-valued predicate given particular
information about the individual case. This calculation is based on the
principle of Maximum Entropy (ME), sometimes called the principle of least
information, and gives the most unbiased probability estimate given the
available evidence. Previous methods for computing maximum entropy values shows
that they are either very restrictive in the probabilistic information
(constraints) they can use or combinatorially explosive. The computational
complexity of the new procedure depends on the inter-connectedness of the
constraints, but in practical cases it is small. In addition, the maximum
entropy method can give a measure of how accurately a calculated conditional
probability is known.

Cited by __201____ __

other theories – imprecise - statistics

+ **Towards a frequentist theory of
upper and lower probability**

P Walley, TL Fine - The Annals of Statistics 10, Number 3 (1982), 741-761

http://projecteuclid.org/euclid.aos/1176345868

We present elements of a frequentist theory of
statistics for concepts of upper and lower (interval-valued) probability (IVP),
defined on finite event algebras. We consider IID models for unlinked
repetitions of experiments described by IVP and suggest several generalizations
of standard notions of independence, asymptotic certainty and estimability.
Instability of relative freqencies is favoured under our IID models. Moreover,
generalizations of Bernoulli's Theorem give some justification for the
estimation of an underlying IVP mechanism from fluctuations of relative
frequencies. Our results indicate that an objectivist, frequency- or
propensity-oriented, view of probability does not necessitate an additive
probability concept, and that IVP models can represent a type of indeterminacy
not captured by additive probability.

applications - diagnostics

+ **Model-based diagnostics and probabilistic assumption-based reasoning**

J Kohlas, B Anrig, R Haenni, PA Monney - Artificial Intelligence 104, Issues
1–2, September 1998, Pages 71–106

http://www.sciencedirect.com/science/article/pii/S0004370298000605

The mathematical foundations of model-based
diagnostics or diagnosis from first principles have been laid by Reiter (1987).
In this paper we extend Reiter's ideas of model-based diagnostics by
introducing probabilities into Reiter's framework. This is done in a
mathematically sound and precise way which allows one to compute the posterior
probability that a certain component is not working correctly given some
observations of the system. A straightforward computation of these
probabilities is not efficient and in this paper we propose a new method to
solve this problem. Our method is logic-based and borrows ideas from
assumption-based reasoning and ATMS. We show how it is possible to determine
arguments in favor of the hypothesis that a certain group of components is not
working correctly. These arguments represent the symbolic or qualitative aspect
of the diagnosis process. Then they are used to derive a quantitative or
numerical aspect represented by the posterior probabilities. Using two new
theorems about the relation between Reiter's notion of conflict and our notion
of argument, we prove that our so-called degree of support is nothing but the
posterior probability that we are looking for. Furthermore, a model where each
component may have more than two different operating modes is discussed and a
new algorithm to compute posterior probabilities in this case is presented.

other theories – association rules

**Association rules and Dempster-Shafer theory of evidence**

T Murai, Y Kudo, Y Sato - Discovery Science, Lecture Notes in Computer
Science Volume 2843, 2003, pp 377-384

http://link.springer.com/chapter/10.1007%2F978-3-540-39644-4_36

The standard definitions of confidence for
association rules was proposed by Agrawal et al. based on the idea that
co-occurrences of items in one transaction are evidence for association between
the items. Since such definition of confidence is nothing but a conditional
probability, even weights are a priori assigned to each transaction that
contains the items in question at the same time. All of such transactions,
however, do not necessarily give us such evidence because some co-occurrences
might be contingent. Thus the D-S theory is introduced to discuss how each
transaction is estimated as evidence.

debate - foundations

+ **Understanding evidential reasoning**

EH Ruspini, JD Lowrance, TM Strat - International Journal of Approximate
Reasoning 6, 401-424, 1992

http://www.sciencedirect.com/science/article/pii/0888613X9290033V

We address recent criticisms of evidential
reasoning, an approach to the analysis of imprecise and uncertain information
that is based on the Dempster-Shafer calculus of evidence.

We show that evidential reasoning can be
interpreted in terms of classical probability theory and that the
Dempster-Shafer calculus of evidence may be considered to be a form of
generalized probabilistic reasoning based on the representation of
probabilistic ignorance by intervals of possible values. In particular, we
emphasize that it is not necessary to resort to nonprobabilistic or
subjectivist explanations to justify the validity of the approach.

We answer conceptual criticisms of evidential
reasoning primarily on the basis of the criticism's confusion between the
current state of development of the theory — mainly theoretical limitations in
the treatment of conditional information — and its potential usefulness in
treating a wide variety of uncertainty analysis problems. Similarly, we
indicate that the supposed lack of decision-support schemes of generalized
probability approaches is not a theoretical handicap but rather an indication
of basic informational shortcomings that is a desirable asset of any formal
approximate reasoning approach. We also point to potential shortcomings of the underlying
representation scheme to treat probabilistic reasoning problems.

We also consider methodological criticisms of the
approach, focusing primarily on the alleged counterintuitive nature of
Dempster's combination formula, showing that such results are the result of its
misapplication. We also address issues of complexity and validity of scope of
the calculus of evidence.

Cited by __40____ __

other theories - GTU

+ **Generalized theory of uncertainty (GTU)—principal concepts and ideas**

LA Zadeh - Computational Statistics & Data Analysis, Volume 51, Issue
1, 1 November 2006, Pages 15–46

http://link.springer.com/chapter/10.1007/3-540-34777-1_1

Uncertainty is an attribute of information. The
path-breaking work of Shannon has led to a universal acceptance of the thesis
that information is statistical in nature. Concomitantly, existing theories of
uncertainty are based on probability theory. The generalized theory of
uncertainty (GTU) departs from existing theories in essential ways. First, the
thesis that information is statistical in nature is replaced by a much more
general thesis that information is a generalized constraint, with statistical
uncertainty being a special, albeit important case. Equating information to a
generalized constraint is the fundamental thesis of GTU. Second, bivalence is
abandoned throughout GTU, and the foundation of GTU is shifted from bivalent
logic to fuzzy logic. As a consequence, in GTU everything is or is allowed to
be a matter of degree or, equivalently, fuzzy. Concomitantly, all variables
are, or are allowed to be granular, with a granule being a clump of values
drawn together by a generalized constraint. And third, one of the principal
objectives of GTU is achievement of NL-capability, that is, the capability to operate
on information described in natural language. NL-capability has high importance
because much of human knowledge, including knowledge about probabilities, is
described in natural language. NL-capability is the focus of attention in the
present paper. The centerpiece of GTU is the concept of a generalized
constraint. The concept of a generalized constraint is motivated by the fact
that most real-world constraints are elastic rather than rigid, and have a
complex structure even when simple in appearance. The paper concludes with
examples of computation with uncertain information described in natural
language.

entropy - specificity

**Entropy and specificity in a mathematical theory of evidence**

RR Yager - Classic Works of the Dempster-Shafer Theory of Belief Functions,
Studies in Fuzziness and Soft Computing Volume 219, 2008, pp 291-310

http://link.springer.com/chapter/10.1007%2F978-3-540-44792-4_11

We review Shafer’s theory of evidence. We then
introduce the concepts of entropy and specificity in the framework of Shafer’s
theory. These become complementary aspects in the indication of the quality of
evidence.

frameworks – inference ?

**- A
calculus for mass assignments in evidential reasoning**

JF Baldwin - Advances in the Dempster-Shafer theory of evidence (RR
Yager; M Fedrizzi; J Kacprzyk Eds.), 513-531, 1994

http://research-information.bristol.ac.uk/en/publications/a-calculus-for-mass-assignments-in-evidential-reasoning%28a2af2a40-b254-428d-b66f-718f8ec8ff61%29.html

no
abstract

**[book] Classic
Works on the Dempster-Shafer Theory of Belief Functions**

L Liu, RR Yager - Studies in Fuzziness and Soft Computing, Springer, 2008

http://www.springer.com/gp/book/9783540253815

machine learning – classification - KNN

+ **A k-nearest neighbor classification rule based on Dempster-Shafer theory**

T Denoeux - Systems, Man and Cybernetics, IEEE Transactions on Systems,
Man and Cybernetics, Vol 25, No 5, pp. 804-813, 1995

In this paper, the problem of classifying an unseen
pattern on the basis of its nearest neighbors in a recorded data set is
addressed from the point of view of Dempster-Shafer theory. Each neighbor of a
sample to be classified is considered as an item of evidence that supports
certain hypotheses regarding the class membership of that pattern. The degree
of support is defined as a function of the distance between the two vectors.
The evidence of the k nearest neighbors is then pooled by means of Dempster's
rule of combination. This approach provides a global treatment of such issues
as ambiguity and distance rejection, and imperfect knowledge regarding the
class membership of training patterns. The effectiveness of this classification
scheme as compared to the voting and distance-weighted k-NN procedures is
demonstrated using several sets of simulated and real-world data.

Cited by __710__

applications – geoscience – fusion - DSmT

+ **Land cover
change prediction with a new theory of plausible and paradoxical reasoning**

S Corgne, L Hubert-Moy, J Dezert, G Mercier - Proceedings of FUSION 2003

The spatial prediction of land cover at the field
scale in winter appears useful for the issue of bare soils reduction in
agricultural intensive regions. High variability of the factors that motivate
the land cover changes between each winter involves integration of uncertainty
in the modelling process. Fusion process wit Dempster-Shafer Theory (DST)
presents some limits in generating errors in decision making when the degree of
conflict, between the sources of evidence that support land cover hypotheses,
becomes important. This paper focuses on the application of Dezert-Smarandache
Theory (DSmT) method to the fusion of multiple land-use attributes for land
cover prediction purpose. Results are discussed and compared with prediction
levels achieved with DST. Through this first application of the
Dezert-Smarandache Theory, we show an example of this new approach ability to
solve some of practical problems where the Dempster-Shafer Theory usually
fails.

decision -
utility

**Linear utility
theory for belief functions**

JY Jaffray -
Operations Research Letters, Volume 8, Issue 2, April 1989, Pages 107–112

http://www.sciencedirect.com/science/article/pii/0167637789900102

In uncertainty situations where knowledge is
described by a Dempster-Shafer belief function (which is more general than a probability measure), von
Neumann-Morgenstern linear utility theory applies and leads to a generalized
expected utility representation of preference which allows for risk-attitude
and ambiguity-attitude (pessimism/optimism).

combination

**On the unicity of Dempster rule of combination**

D Dubois, H Prade - International Journal of Intelligent Systems 1, Issue 2, pages
133–142, Summer 1986

http://onlinelibrary.wiley.com/doi/10.1002/int.4550010204/abstract

Dempster has proposed a rule for the combination of
uncertain items of information issued from several sources. This note proves the
unicity of this rule under an independence assumption. The existence of
alternative rules is stressed, some corresponding to different assumptions,
others pertaining to different types of combination.

applications – data mining

__+ On
subjective measures of interestingness in knowledge discovery__

A Silberschatz, A Tuzhilin – Proc of the First International Conference on Knowledge
Discovery and Data Mining (KDD’95), 1995, pp. 275-281

One
of the central problems in the field of knowledge discovery is the development
of good measures of interestingness of discovered patterns. Such measures of
interestingness are divided into objective measures - those that depend only on
the structure of a pattern and the underlying data used in the discovery
process, and the subjective measures - those that also depend on the class of
users who examine the pattern. The purpose of this paper is to lay the
groundwork for a comprehensive study of subjective measures of interestingness.
In the paper, we classify

these
measures into actionable and unexpected, and examine the relationship between
them. The unexpected measure of interestingness is defined in terms of the
belief system that the user has. Interestingness of a pattern is expressed in
terms of how it affects the belief system.

other theories – fuzzy - decision

+ **Attribute reduction based on generalized fuzzy evidence theory in fuzzy
decision systems**

YQ Yao, JS Mi, ZJ Li - Fuzzy Sets and Systems 170 (2011), 64–75

http://www.sciencedirect.com/science/article/pii/S0165011411000492

Attribute reduction is
viewed as an important issue in data mining and knowledge representation. This
paper studies attribute reduction in fuzzy decision systems based on
generalized fuzzy evidence theory. The definitions of several kinds of
attribute reducts are introduced. The relationships among these reducts are
then investigated. In a fuzzy decision system, it is proved that the concepts
of fuzzy positive region reduct, lower approximation reduct and generalized
fuzzy belief reduct are all equivalent, the concepts of fuzzy upper
approximation reduct and generalized fuzzy plausibility reduct are equivalent,
and a generalized fuzzy plausibility consistent set must be a generalized fuzzy
belief consistent set. In a consistent fuzzy decision system, an attribute set
is a generalized fuzzy belief reduct if and only if it is a generalized fuzzy
plausibility reduct. But in an inconsistent fuzzy decision system, a
generalized fuzzy belief reduct is not a generalized fuzzy plausibility reduct
in general.

applications - manufactoring

M Tabassian, R Ghaderi, R Ebrahimpour - Expert Systems with Applications, Volume 38, Issue 5, May 2011,
Pages 5259–5267

http://www.sciencedirect.com/science/article/pii/S0957417410011760

A new approach for classification of circular knitted fabric
defect is proposed which is based on accepting uncertainty in labels of the
learning data. In the basic classification methodologies it is assumed that
correct labels are assigned to samples and these approaches concentrate on the
strength of categorization. However, there are some classification problems in
which a considerable amount of uncertainty exists in the labels of samples. The
core of innovation in this research has been usage of the uncertain information
of labeling and their combination with the Dempster–Shafer
theory of evidence. The experimental results show the robustness of the
proposed method in comparison with usual classification techniques of
supervised learning where the certain labels are assigned to training data.

combination - frameworks

+ **On Spohn's rule for revision of beliefs**

PP Shenoy - International Journal of Approximate Reasoning 5, Issue 2, March
1991, Pages 149–181

http://www.sciencedirect.com/science/article/pii/0888613X9190035K

The main ingredients of Spohn's theory of epistemic
beliefs are (1) a functional representation of an epistemic state called a
disbelief function and (2) a rule for revising this function in light of new
information. The main contribution of this paper is as follows. First, we
provide a new axiomatic definition of an epistemic state and study some of its
properties. Second, we study some properties of an alternative functional
representation of an epistemic state called a Spohnian belief function. Third,
we state a rule for combining disbelief functions that is mathematically
equivalent to Spohn's belief revision rule. Whereas Spohn's rule is defined in
terms of the initial epistemic state and some features of the final epistemic
state, the rule of combination is defined in terms of the initial epistemic
state and the incremental epistemic state representing the information gained.
Fourth, we state a rule of subtraction that allows one to recover the addendum
epistemic state from the initial and final epistemic states. Fifth, we study
some properties of our rule of combination. One distinct advantage of our rule
of combination is that besides belief revision, it can be used to describe an
initial epistemic state for many variables when this information is given as
several independent epistemic states each involving few variables. Another
advantage of our reformulation is that we can show that Spohn's theory of
epistemic beliefs shares the essential abstract features of probability theory
and the Dempster-Shafer theory of belief functions. One implication of this is
that we have a ready-made algorithm for propagating disbelief functions using
only local computation.

other theories - endorsements

+ **A Framework for Heuristic Reasoning About Uncertainty**

PR Cohen, MR Grinberg – Proceedings of the 8th International Joint Conference on Artificial
Intelligence (IJCAI’83), pp. 355-357, 1983

This paper
describes a theory of reasoning about uncertainly, based on a representation of
states of certainty called endorsements (see Cohen and Grinberg, 1983, for a
more detailed discussion of the theory.) The theory of endorsements is an
alternative to numerical methods for reasoning about uncertainty, such as
subjective Bayesian methods (Shortliffe and Buchanan, 1975; Duda, Hart, and
Nilsson, 1976) and the Shafer-Dempster theory (Shafer, 1976). The fundamental
concern with numerical representations of certainty is that they hide the
reasoning that produces them and thus limit one's reasoning about uncertainty.
While numbers are easy to propagate over inferences, what the numbers mean is
unclear. The theory of endorsements represents the factors that affect
certainty and supports multiple strategies for dealing with uncertainty.

applications - engineering

+ **Uncertainty
quantification of structural response using evidence theory**** (png)**

HR Bae, RV Grandhi, RA Canfield - AIAA Journal, Vol. 41, No. 10 (2003), pp. 2062-2068

Over the past decade, classical probabilistic
analysis has been a popular approach among the uncertainty quantification
methods. As the complexity and performance requirements of a structural system
are increased, the quantification of uncertainty becomes more complicated, and
various forms of uncertainties should be taken into consideration. Because of
the need to characterize the distribution of probability, classical probability
theory may not be suitable for a large complex system such as an aircraft, in
that our information is never complete because of lack of knowledge and
statistical data. Evidence theory, also known as Dempster-Shafer theory, is
proposed to handle the epistemic uncertainty that stems from lack of knowledge
about a structural system. Evidence theory provides us with a useful tool for
aleatory (random) and epistemic (subjective) uncertainties. An intermediate
complexity wing example is used to evaluate the relevance of evidence theory to
an uncertainty quantification problem for the preliminary design of airframe
structures. Also, methods for efficient calculations in large-scale problems
are discussed.

debate - evidence

+ **The concept of distinct evidence**

P Smets – Proceedings of IPMU, 1992

In Dempster-Shafer theory, belief functions induced
by distinct pieces of evidence can be combined by Dempster's rule of
combination. The concept of distinctness has not been formally defined. We
present a tentative definition of the concept of distinctness and compare this
definition with the definition of stochastic independence described in
probability theory.

Cited by __39__

propagation

+ **Representing heuristic knowledge and propagating beliefs in
Dempster-Shafer theory of evidence**

W Liu, JG Hughes, MF McTear - Advances in the Dempster-Shafer theory of
evidence, Pages 441 – 471, 1994

The Dempster-Shafer theory of evidence has been
used intensively to deal with uncertainty in knowledge-based systems. However
the representation of uncertain relationships between evidence and hypothesis
groups (heuristic knowledge) is still a major research problem. This paper
presents an approach to representing such a heuristic knowledge by evidential
mappings which are defined on the basis of mass functions. The relationship
between evidential mappings and multivalued mappings, as well as between
evidential mappings and Bayesian multi-valued causal link models in Bayesian
theory are discussed.

probability transformation

BR Cobb, PP Shenoy - International Journal of Approximate Reasoning 41, Issue 3,
April 2006, Pages 314–330

http://www.sciencedirect.com/science/article/pii/S0888613X05000472

In this paper, we propose the plausibility
transformation method for translating Dempster–Shafer (D–S) belief function
models to probability models, and describe some of its properties. There are
many other transformation methods used in the literature for translating belief
function models to probability models. We argue that the plausibility
transformation method produces probability models that are consistent with D–S
semantics of belief function models, and that, in some examples, the pignistic
transformation method produces results that appear to be inconsistent with
Dempster’s rule of combination.

Cited by __103__

machine learning - classification

NR Pal, S Ghosh – IEEE Transactions on Systems, Man and Cybernetics,
Part A, Vol 31, No 1, pp. 59-66, January 2001

We propose five different ways of integrating
Dempster-Shafer theory of evidence and the rank nearest neighbor classification
rules with a view to exploiting the benefits of both. These algorithms have
been tested on both real and synthetic data sets and compared with the
k-nearest neighbour rule (k-NN), m-multivariate rank nearest neighbour rule
(m-MRNN), and k-nearest neighbour Dempster-Shafer theory rule (k-NNDST), which
is an algorithm that also combines Dempster-Shafer theory with the k-NN rule.
If different features have widely different variances then the distance-based
classifier algorithms like k-NN and k-NNDST may not perform well, but in this
case the proposed algorithms are expected to perform better. Our simulation
results indeed reveal this. Moreover, the proposed algorithms are found to
exhibit significant improvement over the m-MRNN rule.

combination

+ **The consensus operator for combining beliefs**

A Jřsang - Artificial Intelligence 141, Issues 1–2, October 2002, Pages
157–170

http://www.sciencedirect.com/science/article/pii/S000437020200259X

The consensus operator provides a method for
combining possibly conflicting beliefs within the Dempster–Shafer belief
theory, and represents an alternative to the traditional Dempster's rule. This
paper describes how the consensus operator can be applied to dogmatic
conflicting opinions, i.e., when the degree of conflict is very high. It
overcomes shortcomings of Dempster's rule and other operators that have been
proposed for combining possibly conflicting beliefs.

frameworks – generalized evidence theory

**The interpretation of generalized evidence theory**

D Liu, Y Li - Chinese Journal of Computers, 1997

http://cjc.ict.ac.cn/eng/qwjse/view.asp?id=45

This
paper generalizes the concept of random sets, which were used by Dempster, to
Boolean algebra, and discusses the relationship between the uncertainty
structure of information sources and the uncertainty structure of hypothesis
spaces. Generalizing the concepts of upper and lower probabilities, the paper
gives a kind of interpretation for the generalization of evidence theory which
was defined by Guan and Bell, and proves that the conditioning belief functions
defined by Guan and Bell is,in fact, a generalization of Dempster's rule of
condition. The interpretation method, on the one hand, further develops the
generalization of evidence theory, and on the other hand,supplies a feasible
application environment for the generalization of evidence theory.

Machine learning – classification - KNN

+ **An evidence-theoretic k-NN rule with parameter optimization**

LM Zouhal, T Denoeux – IEEE Transactions on Systems, Man, and
Cybernetics Part C, Vol 28, No 2, pp. 263-271, 1998

The paper presents a learning procedure for
optimizing the parameters in the evidence-theoretic k-nearest neighbor rule, a
pattern classification method based on the Dempster-Shafer theory of belief
functions. In this approach, each neighbor of a pattern to be classified is
considered as an item of evidence supporting certain hypotheses concerning the
class membership of that pattern. Based on this evidence, basic belief masses
are assigned to each subset of the set of classes. Such masses are obtained for
each of the k-nearest neighbors of the pattern under consideration and aggregated
using Dempster's rule of combination. In many situations, this method was found
experimentally to yield lower error rates than other methods using the same
information. However, the problem of tuning the parameters of the
classification rule was so far unresolved. The authors determine optimal or
near-optimal parameter values from the data by minimizing an error function.
This refinement of the original method is shown experimentally to result in
substantial improvement of classification accuracy.

debate - foundations

P Cheeseman - Proceedings of the 9th International Joint Conference on
Artificial intelligence (IJCAI’85), Vol 2, pp. 1002-1009

In this paper, it is argued that
probability theory, when used correctly, is suffrcient for the task of
reasoning under uncertainty. Since numerous authors have rejected probability
as inadequate for various reasons, the bulk of the paper is aimed at refuting
these claims and indicating the scources of error. In particular, the
definition of probability as a measure of belief rather than a frequency ratio
is advocated, since a frequency interpretation of probability drastically
restricts the domain of applicability. Other sources of error include the
confusion between relative and absolute probability, the distinction between
probability and the uncertainty of that probability. Also, the interaction of
logic and probability is discusses and it is argued that many extensions of
logic, such as "default logic" are better understood in a
probabilistic framework. The main claim of this paper is that the numerous
schemes for representing and reasoning about uncertainty that have appeared in
the AI literature are unnecessary—probability is all that is needed.

frameworks

+ **A generalization of the Dempster-Shafer theory**

JW Guan, DA Bell – Proceeding of
IJCAI'93, the 13th International Joint Conference on Artifical intelligence, Vol
1, pp. 592-597

The Dempster-Shafer theory gives a solid basis for
reasoning applications characterized by uncertainty. A key feature of the
theory is that propositions are represented as subsets of a set which
represents a hypothesis space. This power set along with the set operations is
a Boolean algebra. Can we generalize the theory to cover arbitrary Boolean
algebras? We show that the answer is yes. The theory then covers, for example,
infinite sets.

The practical advantages of generalization are that
increased flexibility of representation is allowed and that the performance of
evidence accumulation can be enhanced.

In a previous paper we generalized the
Dempster-Shafer orthogonal sum operation to support practical evidence pooling.
In the present paper we provide the theoretical underpinning of that procedure,
by systematically considering familiar evidential functions in turn. For each
we present a "weaker form" and we look at the relationships between
these variations of the functions. The relationships are not so strong as for
the conventional functions. However, when we specialize to the familiar case of
subsets, we do indeed get the wellknown relationships.

hints

**Mathematical foundations of evidence theory**

J Kohlas - Mathematical Models for Handling Partial Knowledge in Artificial
Intelligence, pp. 31-64, 1995

https://www.researchgate.net/publication/246776618_Mathematical_Foundations_of_Evidence_Theory

Reasoning schemes in artificial intelligence (and
elsewhere) use information and knowledge, but the inference my depend on
assumptions which are uncertain. In this case arguments in favour of and
against hypotheses can be derived. These arguments may be weighed by their
likelihoods and thereby the credibility and plausibility of different possible
hypotheses can be obtained. This is, in a nutshell, the idea to be explored and
developed in this article.

applications - fusion

+ **Distributed intrusion detection system based on data fusion method**

Y Wang, H Yang, X Wang and R Zhang - Proc. of the Fifth World Congress
on Intelligent Control and Automation
(WCICA 2004), Volume 5, pp. 4331-4334, 2004

Intrusion detection system (IDS) plays a critical
role in information security because it provides the last line protection for
those protected hosts or networks when intruders elude the first line. In this
paper, we present a novel distributed intrusion detection system, which uses
the Dempster-Shafer's theory of evidence to fuse local information. Our
approach is composed of 2 layers: the lower layer consists of both host and
network based sensors, which are specifically designed to collect local
features and make local decisions to differentiate those easy-to-detect
attacks; the upper layer is a fusion control center, it makes global decisions
on those locally uncertain events by adopting Dempster's combination rule. Our
approach gains the advantages of both host and network based intrusion methods,
and can practice both rule-based and anomaly detection. A simulation is carried
out and result shows that the multi-sensor data fusion model performs much
better than single sensor.

applications – face recognition - image

+ **Human face recognition using Dempster-Shafer theory**

HHS Ip, JMC Ng – Proceedings of ICIP 1994 - DOI:10.1109/ICIP.1994.413578

This paper presents a novel approach to face
recognition based on an application of the theory of evidence (Dempster-Shafer
(1990) theory). Our technique makes use of a set of visual evidence derived
from two projected views (frontal and profile) of the unknown person. The set
of visual evidence and their associate hypotheses are subsequently combined
using the Dempster's rule to output a ranked list of possible candidates. Image
processing techniques developed for the extraction of the set of visual evidence,
the formulation of the face recognition problem within the framework of
Dempster-Shafer theory and the design of suitable mass functions for belief
assignment are discussed. The feasibility of the technique was demonstrated in
an experiment.

applications - engineering

+ **Interpretation of dissolved gas analysis using Dempster-Shafer's theory of evidence**

W Feilhauer, E Handschin – Proc. of the 9^{th} International
Conference on Probabilistic Methods Applied to Power Systems, pp. 1-6, 2006 -
DOI:10.1109/PMAPS.2006.360300

Dissolved gas analysis is a well-known technique to
detect incipient faults of oil-immersed electrical equipment. Interpretation is
usually difficult because there is no unique relationship between the
identified gases and the type of fault. On the basis of IEC standards for the
interpretation of dissolved gas analyses and Dempster-Shafer's theory of
evidence a method to automatically determine the correct diagnosis is
presented. A Markov tree is used for modeling the relationship between the
diagnoses and propagating evidence originating from the measured gas
concentrations. Uncertainty arising from the measurement of the gas
concentration is considered appropriately. The method is verified by
calculating the diagnosis from authentic dissolved gas analyses of several
power transformers and comparing the result to the true fault. Based on the
extent of support for the calculated diagnosis necessary countermeasures are
proposed.

applications - engineering

**A design optimization method using evidence theory**

ZP Mourelatos, J Zhou - J. Mech. Des. 128(4), 901-908 (Dec 28, 2005)

http://mechanicaldesign.asmedigitalcollection.asme.org/article.aspx?articleid=1449054

mourelat@oakland.edu

Early in the engineering design
cycle, it is difficult to quantify product reliability or compliance to
performance targets due to insufficient data or information to model
uncertainties. Probability theory cannot be, therefore, used. Design decisions
are usually based on fuzzy information that is vague, imprecise qualitative,
linguistic or incomplete. Recently, evidence theory has been proposed to handle
uncertainty with limited information as an alternative to probability theory.
In this paper, a computationally efficient design optimization method is
proposed based on evidence theory, which can handle a mixture of epistemic and
random uncertainties. It quickly identifies the vicinity of the optimal point
and the active constraints by moving a hyperellipse in the original design
space, using a reliability-based design optimization (RBDO) algorithm.
Subsequently, a derivative-free optimizer calculates the evidence-based
optimum, starting from the close-by RBDO optimum, considering only the
identified active constraints. The computational cost is kept low by first
moving to the vicinity of the optimum quickly and subsequently using local
surrogate models of the active constraints only. Two examples demonstrate the
proposed evidence-based design optimization method.

other theories - fuzzy

**Generalized probabilities of fuzzy events from fuzzy belief structures**

RR Yager - Information Sciences 28, Issue 1, October 1982, Pages 45–62

http://www.sciencedirect.com/science/article/pii/0020025582900317

We extend Shafer's theory of evidence to include
the ability to have belief structures involving fuzzy sets. We then obtain
under the condition of Bayesian belief structure a whole family of possible
definitions for the probability of fuzzy sets. We also suggest a procedure for
including belief qualification in pruf.

applications - communications

+ **An
evidential model of distributed reputation management**

B Yu, MP Singh - Proceedings of AAMAS’02, July 1519, 2002, Bologna, Italy

byu@eos.ncsu.edu, singh@ncsu.edu

For agents to function effectively in large and
open networks, they must ensure that their *correspondents*, i.e., the
agents they interact with, are trustworthy. Since no central authorities may
exist, the only way agents can find trustworthy correspondents is by
collaborating with others to identify those whose past behavior has been
untrustworthy. In other words, finding trustworthy correspondents reduces to
the problem of distributed reputation management. Our approach adapts the
mathematical theory of evidence to represent and propagate the ratings that
agents give to their correspondents. When evaluating the trustworthiness of a
correspondent, an agent combines its local evidence (based on direct prior
interactions with the correspondent) with the testimonies of other agents
regarding the same correspondent. We experimentally studied this approach to
establish that some important properties of trust are captured by it.

fusion – image

S Le Hégarat-Mascle, I Bloch, D Vidal-Madjar - Pattern Recognition 31,
Issue 11, November 1998, Pages 1811–1823

http://www.sciencedirect.com/science/article/pii/S003132039800051X

Two ways of introducing spatial information in
Dempster–Shafer evidence theory are examined: in the definition of the
monosource mass functions, and, during data fusion. In the latter case, a
“neighborhood” mass function is derived from the label image and combined with
the “radiometric” masses, according to the Dempster orthogonal sum. The main
advantage of such a combination law is to adapt the importance of neighborhood
information to the level of radiometric missing information. The importance of
introducing neighborhood information has been illustrated through the following
application: forest area detection using radar and optical images showing a partial
cloud cover.

other theories – possibility - language

- **On modeling of linguistic information using random sets**

HT Nguyen - Information Sciences 34, Issue 3, December 1984, Pages
265–274

http://www.sciencedirect.com/science/article/pii/0020025584900525

This paper discusses the formal connection between
possibility distributions (Zadeh [21]) and the theory of random sets via
Choquet's theorem. Based upon these relationships, it is suggested that
plausible inferences and modeling of common sense can be derived from the
statistics of random sets. The analysis of subjectivity in meaning
representation of natural languages can be carried out by taking account of
covariates of individuals as in the statistical analysis of survival data.

other theories – incidence - comparison

W Liu, A Bundy - International Journal of Human-Computer Studies 40, No 6,
1009-1032, 1994

http://www.sciencedirect.com/science/article/pii/S1071581984710469

Dealing with uncertainty problems in intelligent
systems has attracted a lot of attention in the AI community. Quite a few
techniques have been proposed. Among them, the Dempster-Shafer theory of
evidence (DS theory) has been widely appreciated. In DS theory, Dempster's
combination rule plays a major role. However, it has been pointed out that the
application domains of the rule are rather limited and the application of the
theory sometimes gives unexpected results. We have previously explored the
problem with Dempster's combination rule and proposed an alternative
combination mechanism in generalized incidence calculus. In this paper we give
a comprehensive comparison between generalized incidence calculus and the
Dempster-Shafer theory of evidence. We first prove that these two theories have
the same ability in representing evidence and combining DS-independent
evidence. We then show that the new approach can deal with some dependent
situations while Dempster's combination rule cannot. Various examples in the
paper show the ways of using generalized incidence calculus in expert systems.

machine learning - relaxation

**Evidence-based pattern-matching relaxation**

P Cucka, A Rosenfeld - Pattern Recognition 26, Issue 9, September 1993,
Pages 1417–1427

http://www.sciencedirect.com/science/article/pii/003132039390147O

In its original form the point pattern-matching
relaxation scheme of Ranade and Rosenfeld did not easily permit the
representation of uncertainty, and it did not exhibit the desirable property
that confidence in consistent pairings of features should increase from one
iteration to the next. Because the process of pooling intrinsic support with
contextual support is essentially a process of evidence combination, it was
suggested by Faugeras that the evidence theory of Dempster and Shafer might be
an appropriate framework for relaxation labeling. Some of the issues involved
in the implementation of such an approach are addressed and results from the
domain of object recognition in SAR imagery are presented.

multicriteria decision

MA Boujelben, YD Smet, A Frikha and H Chabchoub - International Journal of Approximate
Reasoning 50, Issue 8, September 2009, Pages 1259–1278

http://www.sciencedirect.com/science/article/pii/S0888613X09001054

We consider multicriteria decision problems where
the actions are evaluated on a set of ordinal criteria. The evaluation of each
alternative with respect to each criterion may be uncertain and/or imprecise
and is provided by one or several experts. We model this evaluation as a basic
belief assignment (BBA). In order to compare the different pairs of
alternatives according to each criterion, the concept of first belief dominance
is proposed. Additionally, criteria weights are also expressed by means of a
BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical
example.

independence - conditioning

+ **Belief function independence: II. The conditional case**

B Ben Yaghlane, P Smets, K Mellouli - International Journal of
Approximate Reasoning 31, Issues 1–2, October 2002, Pages 31–75

http://www.sciencedirect.com/science/article/pii/S0888613X02000725

In the companion paper [Int. J. Approx. Reasoning
29 (1) (2002) 47], we have emphasized the distinction between non-interactivity
and doxastic independence in the context of the transferable belief model. The
first corresponds to decomposition of the belief function, whereas the second
is defined as irrelevance preserved under Dempster’s rule of combination. We
had shown that the two concepts are equivalent in the marginal case. We proceed
here with the conditional case. We show how the definitions generalize
themselves, and that we still have the equivalence between conditional
non-interactivity and conditional doxastic independence.

applications - vision

+ **A skin detection approach based on the Dempster–Shafer theory of evidence**

M Shoyaib, M Abdullah-Al-Wadud, O Chae - International Journal of Approximate Reasoning 53, Issue 4,
June 2012, Pages 636–659

http://www.sciencedirect.com/science/article/pii/S0888613X12000047

Skin detection is an important step for a wide
range of research related to computer vision and image processing and several
methods have already been proposed to solve this problem. However, most of
these methods suffer from accuracy and reliability problems when they are
applied to a variety of images obtained under different conditions. Performance
degrades further when fewer training data are available. Besides these issues,
some methods require long training times and a significant amount of parameter
tuning. Furthermore, most state-of-the-art methods incorporate one or more thresholds,
and it is difficult to determine accurate threshold settings to obtain
desirable performance. These problems arise mostly because the available
training data for skin detection are imprecise and incomplete, which leads to
uncertainty in classification. This requires a robust fusion framework to
combine available information sources with some degree of certainty. This paper
addresses these issues by proposing a fusion-based method termed
Dempster–Shafer-based Skin Detection (DSSD). This method uses six prominent
skin detection criteria as sources of information (SoI), quantifies their
reliabilities (confidences), and then combines their confidences based on the
Dempster–Shafer Theory (DST) of evidence. We use the DST as it offers a
powerful and flexible framework for representing and handling uncertainties in
available information and thus helps to overcome the limitations of the current
state-of-the-art methods. We have verified this method on a large dataset
containing a variety of images, and achieved a 90.17% correct detection rate
(CDR). We also demonstrate how DSSD can be used when very little training data
are available, achieving a CDR as high as 87.47% while the best result achieved
by a Bayesian classifier is only 68.81% on the same dataset. Finally, a
generalized DSSD (GDSSD) is proposed achieving 91.12% CDR.

foundations - uncertainty

D Kahneman, A Tversky – Cognition 11, Issue 2, March 1982, Pages 143–157

http://www.sciencedirect.com/science/article/pii/0010027782900233

In
contrast to formal theories of judgement and decision, which employ a single
notion of probability, psychological analyses of responses to uncertainty
reveal a wide variety of processes and experiences, which may follow different
rules. Elementary forms of expectation and surprise in perception are reviewed.
A phenomenological analysis is described, which distinguishes external
attributions of uncertainty (disposition) from internal attributions of
uncertainty (ignorance). Assessments of uncertainty can be made in different
modes, by focusing on frequencies, propensities, the strength of arguments, or
direct experiences of confidence. These variants of uncertainty are associated
with different expressions in natural language; they are also suggestive of
competing philosophical interpretations of probability.

other theories – uncertainty - fuzzy

**Toward a general theory of reasoning with uncertainty. I: Nonspecificity
and fuzziness**

RR Yager - International Journal of Intelligent Systems 1, Issue 1, pages
45–67, Spring 1986

http://onlinelibrary.wiley.com/doi/10.1002/int.4550010106/abstract

We described three theories of approximate
reasoning and mathematical evidence. We show that in the face of possibilistic
uncertainty they lead to equivalent inferences. After appropriately extending
the mathematical theory of evidence to the fuzzy environment we show that these
two theories are equivalent in the face of fuzzy and possibilistic uncertainty.

computation - sampling

JC Helton, JD Johnson, WL Oberkampf and CB Storlie - Computer Methods in
Applied Mechanics and Engineering 196, Issues 37–40, 1 August 2007, Pages
3980–3998

http://www.sciencedirect.com/science/article/pii/S004578250700120X

jchelto@sandia.gov

Evidence theory provides an alternative to probability theory
for the representation of epistemic uncertainty in model predictions that
derives from epistemic uncertainty in model inputs, where the descriptor
epistemic is used to indicate uncertainty that derives from a lack of knowledge
with respect to the appropriate values to use for various inputs to the model.
The potential benefit, and hence appeal, of evidence theory is that it allows a
less restrictive specification of uncertainty than is possible within the
axiomatic structure on which probability theory is based. Unfortunately, the
propagation of an evidence theory representation for uncertainty through a
model is more computationally demanding than the propagation of a probabilistic
representation for uncertainty, with this difficulty constituting a serious
obstacle to the use of evidence theory in the representation of uncertainty in
predictions obtained from computationally intensive models. This presentation
describes and illustrates a sampling-based computational strategy for the
representation of epistemic uncertainty in model predictions with evidence
theory. Preliminary trials indicate that the presented strategy can be used to
propagate uncertainty representations based on evidence theory in analysis
situations where naďve sampling-based (i.e., unsophisticated Monte Carlo)
procedures are impracticable due to computational cost.

probability transformation – pignistic - TBM

+ **Constructing the Pignistic Probability Function in a Context of
Uncertainty**

P Smets – Proceedings of Uncertainty in Artificial Intelligence (UAI’89),
pp 29-40, 1989

Many new models have been proposed to quantify
uncertainty. But usually they don't explain how decisions must be derived. In
probability theory, the expected utility model is well established and strongly
justified. We show that such expected utility model can be derived in the other
models proposed to quantify someone's belief. The justification is based on
special bets and some coherence requirements that lead to the derivation of the
so-called generalized insufficient reason principle. In Smets (1988b, 1988c,
1989) we emphasize the existence of two levels where beliefs manifest
themselves: the credal level where beliefs are entertained and the pignistic
level where beliefs are used to take decisions (pignus = a bet in Latin, Smith
1961).

applications -
engineering

**+ Application**

Parikh, CR, Pont, MJ, and Jones - Pattern
Recognition Letters 22, 777–785, 2001

http://www.sciencedirect.com/science/article/pii/S0167865501000149

M.Pont@leicester.ac.uk

This paper is concerned with the
use of Dempster-Shafer theory in ‘fusion’ classifiers. We argue that the use of
predictive accuracy for basic probability assignments can improve the overall
system performance when compared to ‘traditional’ mass assignment techniques.
We demonstrate the effectiveness of this approach in a case study involving the
detection of static thermostatic valve faults in a diesel engine cooling
system.

__Cited by 103__

applications – fusion – activity recognition

J Liao, Y Bi, C Nugent – IEEE Transactions on Information Technology in Biomedicine 15,
No 1, 74-82, 2011

liao-j1@email.ulster.ac.uk

This paper explores a sensor fusion method applied
within smart homes used for the purposes of monitoring human activities in
addition to managing uncertainty in sensor-based readings. A three-layer
lattice structure has been proposed, which can be used to combine the mass
functions derived from sensors along with sensor context. The proposed model
can be used to infer activities. Following evaluation of the proposed
methodology it has been demonstrated that the Dempster-Shafer theory of
evidence can incorporate the uncertainty derived from the sensor errors and the
sensor context and subsequently infer the activity using the proposed lattice
structure. The results from this study show that this method can detect a
toileting activity within a smart home environment with an accuracy of 88.2%.

applications – fusion - military

+ **Multisensor fusion in the frame of evidence theory for landmines
detection**

S Perrin, E Duflos, P Vanheeghe and A Bibaut - IEEE Transactions on Systems, Man, and
Cybernetics part C 34, No 2, 485-498, 2004

In the frame of humanitarian antipersonnel mines
detection, a multisensor fusion method using the Dempster-Shafer evidence
theory is presented. The multisensor system consists of two sensors-a ground
penetrating radar (GPR) and a metal detector (MD). For each sensor, a new
features extraction method is presented. The method for the GPR is mainly based
on wavelets and contours extraction. First simulations on a limited set of data
show that an improvement in detection and false alarms rejection, for the GPR
as a standalone sensor, could be obtained. The MD features extraction method is
mainly based on contours extraction. All of these features are then fused with
the GPR ones in some specific cases in order to determine a new feature. From
these results, belief functions, as defined in the evidence theory, are then
determined and combined thanks to the orthogonal sum. First results in terms of
detection and false alarm rates are presented for a limited set of real data
and a comparison is made between the two cases: with or without multisensor
fusion.

approximation – computation - transformation

**Analyzing approximation algorithms in the theory of evidence**

AL Jousselme, D Grenier and E Bosse - Proc. SPIE 4731, Sensor Fusion:
Architectures, Algorithms, and Applications VI, 65 (March 8, 2002);
doi:10.1117/12.458371

http://proceedings.spiedigitallibrary.org/proceeding.aspx?articleid=887318

The major drawback of the Dempster-Shafer's theory
of evidence is its computational burden. Indeed, the Dempster's rule of
combination involves an exponential number of focal elements, that can be
unmanageable in many applications. To avoid this problem, some approximation
rules or algorithms have been explored for both reducing the number of focal
elements and keeping a maximum of information in the next belief function to be
combined. Some studies have yet to be done which compare approximation
algorithms. The criteria used always involve pignistic transformations, and by
that a loss of information in both the original belief function and the
approximated one. In this paper, we propose to analyze some approximation
methods by computing the distance between the original belief function and the
approximated one. This real distance allows then to quantify the quality of the
approximation. We also compare this criterion to other error criteria, often
based on pignistic transformations. We show results of Monte-Carlo simulations,
and also of an application of target identification.

applications – medical - EEG

+ **EMG pattern recognition based on artificial intelligence techniques**

SH Park, SP Lee - IEEE Transactions on Rehabilitation Engineering 6, No 4,
400-405, 2002

This
paper presents an electromyographic (EMG) pattern recognition method to
identify motion commands for the control of a prosthetic arm by evidence
accumulation based on artificial intelligence with multiple parameters. The
integral absolute value, variance, autoregressive (AR) model coefficients, linear
cepstrum coefficients, and adaptive cepstrum vector are extracted as feature
parameters from several time segments of EMG signals. Pattern recognition is
carried out through the evidence accumulation procedure using the distances
measured with reference parameters. A fuzzy mapping function is designed to
transform the distances for the application of the evidence accumulation
method. Results are presented to support the feasibility of the suggested
approach for EMG pattern recognition.

machine learning – decision trees

**Multiple binary decision tree classifiers**

S Shlien - Pattern Recognition 23, Issue 7, 1990, Pages 757–763

http://www.sciencedirect.com/science/article/pii/0031320390900986

Binary decision trees based on nonparametric
statistical models of the data provide a solution to difficult decision problems
where there are many classes and many available features related in a complex
manner. Unfortunately, the technique requires a very large training set and is
often limited by the size of the training set rather than by the discriminatory
power of the features. This paper demonstrates that higher classification
accuracies can be obtained from the same training set by using a combination of
decision trees and by reaching a consensus using Dempster and Shafer's theory
of evidence.

combination – graphical models - computation

+ **Dempster's rule for evidence ordered in a complete directed acyclic
graph**

U Bergsten, J Schubert - International Journal of Approximate Reasoning 9, Issue 1,
August 1993, Pages 37–73

For the case of evidence ordered in a complete
directed acyclic graph this paper presents a new algorithm with lower
computational complexity for Dempster's rule than that of step-by-step
application of Dempster's rule. In this problem, every original pair of
evidences, has a corresponding evidence against the simultaneous belief in both
propositions. In this case, it is uncertain whether the propositions of any two
evidences are in logical conflict. The original evidences are associated with
the vertices and the additional evidences are associated with the edges. The
original evidences are ordered, i.e., for every pair of evidences it is
determinable which of the two evidences is the earlier one. We are interested
in finding the most probable completely specified path through the graph, where
transitions are possible only from lower- to higher-ranked vertices. The path
is here a representation for a sequence of states, for instance a sequence of
snapshots of a physical object's track. A completely specified path means that
the path includes no other vertices than those stated in the path
representation, as opposed to an incompletely specified path that may also
include other vertices than those stated. In a hierarchical network of all
subsets of the frame, i.e., of all incompletely specified paths, the original
and additional evidences support subsets that are not disjoint, thus it is not
possible to prune the network to a tree. Instead of propagating belief, the new
algorithm reasons about the logical conditions of a completely specified path
through the graph. The new algorithm is O(|Θ| log|Θ|), compared to
O(|Θ|^{log|Θ|}) of the classic brute force algorithm. After a
detailed presentation of the reasoning behind the new algorithm we conclude
that it is feasible to reason without approximation about completely specified
paths through a complete directed acyclic graph.

frameworks – applications – vision - computation

+ **Plausible
reasoning and the theory of
evidence**

G Reynolds, D Strahman, N Lehrer, L Kitchen – COINS Technical Report
86-11, University of Massachusetts Amherst, MA, USA, 1986

https://web.cs.umass.edu/publication/details.php?id=10

In this paper we describe the mathematical foundations of a
knowledge representation and evidence combination framework and relate it to
the theory of evidential reasoning as developed by dempster and shafer.
Although our discussion takes place in the context of computer vision, the
results are applicable to problems in knowledge representation and data
interpretation. Our representation, called pl-functions, and a simple
multiplicative combination rule is shown to be equivalent to a sub-class of the
family of mass-functions as described by shafer with dempster''s rule as the
combination function. However, the simpler combination rule has a complexity
which is linear with respect to the number of elements in the frame of discernment.
This is a tremendous computational advantage over the general theory which
provides a combination rule exponential with respect to the number of objects
over which we are reasoning. We also discuss a method which allows our
representation to be automatically generated from statistical data.

debate - foundations

__A simple
view of the Dempster-Shafer theory
of evidence__

LA Zadeh – Berkeley Cognitive Science, Rep. No. 27 Univ. of California,
Berkeley, 1984

no
abstract (see Zadeh 1986, AI Magazine)

other theories

+ **A
mathematical theory of
evidence for GLS Shackle**

G Fioretti - Mind & Society 2, Issue 1, pp 77-98, 2001

http://link.springer.com/article/10.1007%2FBF02512076

Evidence Theory is a branch of mathematics that
concerns combination of empirical evidence in an individual’s mind in order to
construct a coherent picture of reality. Designed to deal with unexpected
empirical evidence suggesting new possibilities, evidence theory is compatible
with Shackle’s idea of decision-making as a creative act. This essay
investigates this connection in detail, pointing to the usefulness of evidence
theory to formalise and extend Shackle’s decision theory. In order to ease a
proper framing of the issues involved, evidence theory is compared with
sub-additive probability theory and Ewens’s infinite alleles model.
Furthermore, the original version of evidence theory is presented along with
its most recent developments.

**[book] Advances
in Dempster-Shafer theory of
evidence**

J Kacprzyk, M Fedrizzi (editors) - Wiley, New York, 1994

applications – activity recognition

J Liao, Y Bi, C Nugent – Proc of Intelligent Environments (IE), 46-51, 2010 – DOI: 10.1109/IE.2010.16

This paper explores an improvement to activity recognition
within a Smart Home environment using the Dempster-Shafer theory of evidence.
This approach has the ability to be used to monitor human activities in
addition to managing uncertainty in sensor based readings. A three layer
lattice structure has been proposed, which can be used to combine the mass
functions derived from sensors along with sensor context and subsequently can
be used to infer activities. From the total 209 recorded activities throughout
a two week period, 85 toileting activities were considered. The results from
this work demonstrated that this method was capable of detecting 75 of the
toileting activities correctly within a Smart Home environment equating to a
classification accuracy of 88.2%.

other theories – rough sets

+ **Rough set theory and its applications to data analysis**

Z Pawlak - Cybernetics & Systems 29, Issue 7, 661-688, 1998

This paper gives basic ideas of rough set theory a new
approach to data analysis. The lower and upper approximation of a set, the
basic operations of the theory, are intuitively explained and formally defined.
Some applications of rough set theory are briefly outlined and some future
problems are outlined.

__Cited by 1030__

applications - reliability

**+ ****Application of Dempster–Shafer theory in condition monitoring
applications: a case study**

CR Parikh, MJ Pont, N Barrie Jones - Pattern Recognition Letters 22,
Issues 6–7, May 2001, Pages 777–785

http://www.sciencedirect.com/science/article/pii/S0167865501000149

This paper is concerned with the use of
Dempster-Shafer theory in ‘fusion’ classifiers. We argue that the use of predictive
accuracy for basic probability assignments can improve the overall system
performance when compared to ‘traditional’ mass assignment techniques. We
demonstrate the effectiveness of this approach in a case study involving the
detection of static thermostatic valve faults in a diesel engine cooling
system.

debate – foundations – compatibility relation

+ **Non-monotonic compatibility relations in the theory of evidence**

RR Yager - International Journal of Man-Machine Studies 29, Issue 5, 1988,
Pages 517–537

http://www.sciencedirect.com/science/article/pii/S0020737388800105

A belief structure, *m*, provides a generalized format for representing uncertain
knowledge about a variable. We suggest that the idea of one belief structure
being more specific than another is related to the plausibility-certainty
interval, more fundamentally, how well we know the probability structure. A
compatibility relation provides a structure for obtaining information about one
variable based upon a second variable. An inference scheme in the theory of
evidence concerns itself with the use of a compatibility relation and a belief
structure on one variable to infer a belief structure on the second variable.
The problem of monotonicity in this situation can be related to change in the
specificity of the inferred belief structure as the antecedent belief structure
becomes more specific. We show that the usual compatibility relations, type I,
are always monotonic. We introduce type II compatibility relations and show
that a special class of these, which we call irregular, are needed to represent
non-monotonic relations between variables. We discuss a special class of non-monotonic
relations called default relations.

other theories - fuzzy

**On the concept of possibility-probability consistency**

M Delgado, S Moral - Fuzzy Sets and Systems 21, Issue 3, March 1987, Pages 311–318

http://www.sciencedirect.com/science/article/pii/0165011487901321

Both probability and
possibility may be seen as information about an experiment. It is conceivable
to have at some time these two forms of information about a same experiment and
then the question of the relation between them arises at once. In this paper
some aspects of the concept of possibility-probability consistency are studied.
The consistency is considered as a fuzzy property relative to the coherence
between possibilistic and probabilistic information. We analyse several
measures of the degree of consistency and introduce an axiomatic to
characterize them.

propagation – other theories - comparison

+ **Equivalence of methods for uncertainty propagation of real-valued random
variables**

HM Regan, S Ferson, D Berleant - International Journal of Approximate Reasoning 36, Issue 1,
April 2004, Pages 1–30

http://www.sciencedirect.com/science/article/pii/S0888613X03001245

In this paper we
compare four methods for the reliable propagation of uncertainty through
calculations involving the binary operations of addition, multiplication,
subtraction and division. The methods we investigate are: (i) dependency bounds
convolution; (ii) Distribution Envelope Determination; (iii) interval
probabilities; and (iv) Dempster–Shafer belief functions. We show that although
each of these methods were constructed for different types of applications,
they converge to equivalent methods when they are restricted to cumulative
distribution functions on the positive reals. We also show that while some of
the methods have been formally constructed to deal only with operations on
random variables under an assumption of independence, all of the methods can be
extended to deal with unknown dependencies and perfect positive and negative
dependence among variables.

other theories - capacities

**Decomposable capacities, distorted probabilities and concave capacities**

A Chateauneuf - Mathematical Social Sciences 31, Issue 1, February 1996,
Pages 19–37

http://www.sciencedirect.com/science/article/pii/0165489695007946

During the last few
years, capacities have been used extensively to model attitudes towards
uncertainty. We describe the links between some classes of capacities, namely
between decomposable capacities introduced by Dubois and Prade and other
capacities, such as concave or convex capacities, and distorted probabilities
that appeared in two new models of non-additive expected utility theory
(Schmeidler,*Econometrica*,
1989, 57, 571–587; Yaari, *Econometrica*,
1987, 55, 95–115). It is shown that the most well-known decomposable capacities
prove to be distorted probabilities, and that any concave distortion of a
probability is decomposable. The paper ends by successively characterizing
decomposable capacities that are concave distortions of probabilities, and ⊥-decomposable capacities (for triangular conorms ⊥) that are concave, since decomposable capacities prove to be much more
related to concavity than convexity.

applications - agents

+ **Situation awareness in intelligent agents: Foundations for a theory of
proactive agent behavior**

R So, L Sonenberg - Proceedings of IAT 2004,
86-92, 2004

This work outlines a
computational model of situation awareness. This model serves two purposes.
First, it provides a detailed description for our everyday notion of situation
awareness. Second, it offers an alternative perspective of looking into the
nature and characteristics of proactive behavior in intelligent agent systems.
Most of the existing definitions of proactiveness do not possess the relevant
details to encourage rigorous studies of enhancing an agent's capability to
anticipate and deal with foreseeable situations. This work intends to fill this
gap by suggesting a working definition of proactiveness, which place additional
emphasis on its underpinning cognitive process and its role in directing an
agent's attention. A brief discussion of using situation awareness as
meta-level control to direct a resource-bounded agent's attention is also
included.

other theories - interval

**Interval probability theory for evidential support**

W Cui, DI Blockley - International Journal of Intelligent Systems 5,
Issue 2, pages 183–192, June 1990

onlinelibrary.wiley.com/doi/10.1002/int.4550050204/

An
interval theory of probability is presented for use as a measure of evidential
support in knowledge-based systems. an interval number is used to capture, in a
relatively simple manner, features of fuzziness and incompleteness. the vertex
method is used for the interval analysis. A new parameter (also an interval
number), p, called the degree of dependence is introduced. the relationship of
this interval probability with the theories of Dempster-Shafer, fuzzy sets, and
Baldwin's support logic are discussed. the advantage of the theory is that it
is based on a development of the axioms of probability, but allows that
evidential support for a conjecture be separated from evidential support for
the negation of the conjecture.

Cited by __81____ __

applications – reliability - engineering

BS Yang, KJ Kim - Mechanical Systems and Signal Processing 20, Issue 2,
February 2006, Pages 403–420

http://www.sciencedirect.com/science/article/pii/S0888327004001694

This paper presents an approach for the fault diagnosis in
induction motors by using Dempster–Shafer theory. Features are extracted from
motor stator current and vibration signals and with reducing data transfers.
The technique makes it possible for on-line application. Neural network is
trained and tested by the selected features of the measured data. The fusion of
classification results from vibration and current classifiers increases the
diagnostic accuracy. The efficiency of the proposed system is demonstrated by
detecting motor electrical and mechanical faults originated from the induction
motors. The results of the test confirm that the proposed system has potential
for real-time applications.

applications – retrieval - empirical

+ **Experimenting
on Dempster-Shafer's theory of
evidence in information retrieval**

I Ruthven, M Lalmas - Journal of Intelligent Information Systems, 1997

igr, mounia@dcs.gla.ac.uk

This report describes a set of experiments
investigating the use of Dempster-Shafer's Theory of Evidence in Information
Retrieval. Our experiments use various indexing and retrieval methods to
exploit Dempster-Shafer’s theory and we outline the reasons for the success or
failure of the different approaches taken.

applications - retrieval

**Information retrieval and Dempster-Shafer's theory of evidence**

M Lalmas - Applications of Uncertainty Formalisms, Lecture Notes in
Computer Science Volume 1455, 1998, pp 157-176

http://link.springer.com/chapter/10.1007%2F3-540-49426-X_8

This paper describes the use of the Dempster-Shafer
theory of evidence to construct an information retrieval model that aims to
capture four essential features of information: structure, significance,
uncertainty and partiality. We show that Dempster-Shafer’s initial framework
allows the representation of the structure and the significance of information,
and that the notion of refinement later introduced by Shafer allows the
representation of the uncertainty and the partiality of information. An
implementation of the model is briefly discussed.

frameworks -
hints

+ **Representation of
evidence by hints**

J Kohlas, PA Monney – Classic Works of the Dempster-Shafer
Theory of Belief Functions, Studies in Fuzziness and Soft Computing Volume 219,
2008, pp 665-681- DOI: 10.1007/978-3-540-44792-4_26

This paper introduces a mathematical model of
a hint as a body of imprecise and uncertain information. Hints are used to
judge hypotheses: the degree to which a hint supports a hypothesis and the
degree to which a hypothesis appears as plausible in the light of a hint are
defined. This leads in turn to support- and plausibility functions. Those
functions are characterized as set functions which are normalized and monotone
or alternating of order 1. This relates the present work to G. Shafer's
mathematical theory of evidence. However, whereas Shafer starts out with an
axiomatic definition of belief functions, the notion of a hint is considered
here as the basic element of the theory. It is shown that a hint contains more
information than is conveyed by its support function alone. Also hints allow
for a straightforward and logical derivation of Dempster's rule for combining
independent and dependent bodies of information. This paper presents the
mathematical theory of evidence for general, infinite frames of discernment
from the point of view of a theory of hints.

decision

+ **Attribute reduction based on evidence theory in incomplete decision
systems**

W-Z Wu - Information Sciences 178, Issue 5, 1 March 2008, Pages 1355–1371

http://www.sciencedirect.com/science/article/pii/S0020025507004902

wuwz@zjou.edu.cn

Attribute reduction is a basic issue in knowledge
representation and data mining. This paper deals with attribute reduction in
incomplete information systems and incomplete decision systems based on
Dempster–Shafer theory of evidence. The concepts of plausibility reduct and
belief reduct in incomplete information systems as well as relative
plausibility reduct and relative belief reduct in incomplete decision systems
are introduced. It is shown that in an incomplete information system an
attribute set is a belief reduct if and only if it is a classical reduct and a
plausibility consistent set must be a classical consistent set. In a consistent
incomplete decision system, the concepts of relative reduct, relative
plausibility reduct, and relative belief reduct are all equivalent. In an
inconsistent incomplete decision system, an attribute set is a relative
plausibility reduct if and only if it is a relative reduct, a plausibility
consistent set must be a belief consistent set, and a belief consistent set is
not a plausibility consistent set in general.

combination – logic - expert

+ **Syllogistic reasoning as a basis for combination of evidence in expert
systems**

LA Zadeh - Proceedings of IJCAI'85, Volume 1, pp 417-419, 1985 -
ISBN:0-934613-02-8

In the presence of uncertainty, computation of the certainty factor of a hypothesis requires, in general, the availability of rules for combining evidence under chaining, disjunction and conjunction. The method described in this paper is based on the use of what may be viewed as a generalization of syllogistic reasoning in classical logic--a generalization in which numerical or, more generally, fuzzy quantifiers assume the