[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 13532

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

R YagerM FedrizziJ Kacprzyk (editors) – John Wiley, 1994

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.”

Cited by 455



+ Review of a mathematical theory of evidence

LA Zadeh - AI magazine, Vol.5, No. 3, 1984 - DOI:

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.

Cited by 423


+ A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination

LA Zadeh - AI magazine, Vol. 7, No. 2, 1986 - DOI:

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.

Cited by 553 


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

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.


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.

Cited by 278



+ 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.

Cited by 471



Approximations for efficient computation in the theory of evidence

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

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.

Cited by 194

advances – entropy - information

+ Entropy and specificity in a mathematical theory of evidence

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

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.

Cited by 349

computation - approximation

+ Approximation algorithms and decision making in the Dempster-Shafer theory of evidence—an empirical study

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.

Cited by 147


+ 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.

Cited by 263


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).

Cited by 279

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.

Cited by 169

debate - foundations

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

JY HalpernR 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 HalpernR 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.



+ Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence

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

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.

Cited by 257

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

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.

Cited by 433

Conflict - combination

+ Analyzing the degree of conflict among belief functions

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

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.

Cited by 253


+ 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.

Cited by 316

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.

Cited by 303


+ On the justification of Dempster's rule of combination

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

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.

Cited by 228

advances - information

- Properties of measures of information in evidence and possibility theories

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

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.

Cited by 147

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.

Cited by 154


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

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.

Cited by 171

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

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.

Cited by 343


other theories – rough sets

+ Interpretations of belief functions in the theory of rough sets

YY YaoPJ 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.

Cited by 144


machine learning - classification

Combining the classification results of independent classifiers based on Dempster/Shafer theory of evidence

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

no abstract

Cited by 137


+ 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

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.

Cited by 87

other theories – possibility - logic

+ Possibilistic logic

D DuboisJ LangH 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.

Cited by 853


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

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.

Cited by 228

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-AniM 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.

Cited by 149

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 (XY) and in TMC, the distribution of the pair (XY) is the marginal distribution of a Markov process (XUY), 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


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

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

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 m1, …, mn over a frame of discernment Θ, and a set A  Θ, the problem of computing the combined basic probability value (m1  mn)(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 (Bel1  Beln(A), (Pl1  Pln)(A), and (Q1  Qn)(A) are #P-complete.

Cited by 104

consonant approximation

+ Consonant approximations of belief functions

D DuboisH 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.

Cited by 197

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


+ Reasoning with imprecise belief structures

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

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.

Cited by 137


- On the evidence inference theory

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

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.

Cited by 45

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

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.

Cited by 86

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

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.

Cited by 486

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

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

+ Linguistic aggregation operators for linguistic decision making based on the Dempster-Shafer theory of evidence

JM MerigóM CasanovasL Martínez - International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Volume 18, Issue 03, page 287, June 2010

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.

Cited by 77


+ Dempster-Shafer theory

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

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

no abstract

Cited by 66

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.

Cited by 64

Machine learning – classification – applications - medical

+ Some aspects of Dempster-Shafer evidence theory for classification of multi-modality medical images taking partial volume effect into account

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.

Cited by 218

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

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.

Cited by 556


+ A new combination of evidence based on compromise

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

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.

Cited by 54

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

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.

Cited by 44

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.

Cited by 46

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

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 Systems1(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).

Cited by 48

other theories – rough sets

+ Rough mereology: A new paradigm for approximate reasoning

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

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.

Cited by 416

applications - measurement

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

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

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.

Cited by 77

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

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.

Cited by 36

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.

Cited by 45

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

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.

Cited by 221

applications – information retrieval

[thesis] Theories of information and uncertainty for the modelling of information retrieval: an application of situation theory and Dempster-Schafer's theory of evidence

M Lalmas - University of Glasgow, 1996

Cited by 41


other theories – fuzzy - advances

+ Modeling vague beliefs using fuzzy-valued belief structures

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

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.

Cited by 111

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

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

+ Representing and retrieving structured documents using the Dempster-Shafer theory of evidence: Modelling and evaluation

M LalmasI Ruthven - Journal of Documentation 54, No 5, 1998

In this paper we report on a theoretical model of structured document indexing and retrieval based on the DempsterShafer 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.

Cited by 57

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

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

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

Cited by 32

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

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.

Cited by 101

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

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.

Cited by 137

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

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

Theory of evidence - A survey of its mathematical foundations, applications and computational aspects

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

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.

Cited by 45

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

Cited by 37


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

no abstract

Cited by 92


machine learning - classification

+ Handling possibilistic labels in pattern classification using evidential reasoning

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

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.

Cited by 100

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

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.

Cited by 28

applications – documents – information retrieval

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

I RuthvenM Lalmas - Journal of Intelligent Information Systems, Volume 19Issue 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.

Cited by 44

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

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 ak-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.

Cited by 234

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

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).

Cited by 33


Classic works of the Dempster-Shafer theory of belief functions

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

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.

Cited by 150

other theories – capacities – moebius - geometry

- Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion

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

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 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

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.

Cited by 49

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

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.

Cited by 36

debate – belief update

+ Evidence, knowledge, and belief functions

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

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.

Cited by 59

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

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.

Cited by 49

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

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.

Cited by 104

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.

Cited by 45


applications – sensor fusion - aerospace

Methods for multisensor classification of airborne targets integrating evidence theory

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

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.

Cited by 23

other theories – frameworks – probabilistic logic

+ A logic for reasoning about probabilities

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

We consider a language for reasoning about probability which allows us to make statements such as “the probability of E1 is less than 1/3” and “the probability of E1 is at least twice the probability of E2,” where E1and E2 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.

Cited by 581

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

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.

Cited by 31

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

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.

Cited by 32

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

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(N2).

Cited by 71

applications – measurement – random fuzzy variables

+ Modeling and processing measurement uncertainty within the theory of evidence: Mathematics of random–fuzzy variables

A FerreroS 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.

Cited by 33

Applications – ubiquitous computing

+ Using Dempster-Shafer theory of evidence for situation inference

S McKeever, J YeL CoyleS 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.

Cited by 32

applications – classification – remote sensing – earth sciences

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

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

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.

Cited by 38


+ Epistemic logics, probability, and the calculus of evidence

EH Ruspini - Proceedings of the 10th 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.

Cited by 73

applications - communications

+ An enhanced cooperative spectrum sensing scheme based on evidence theory and reliability source evaluation in cognitive radio context

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

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.

Cited by 224

applications - engineering

Investigation of evidence theory for engineering applications

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

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 Rth level ranking statistics. Rth 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.

Cited by 32

computation – approximation - combination

+ Approximating the combination of belief functions using the fast Moebius transform in a coarsened frame

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

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.

Cited by 67

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

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.

Cited by 38

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

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.

Cited by 403

applications – civil engineering

+ Estimating risk of contaminant intrusion in water distribution networks using Dempster–Shafer theory of evidence

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.

Cited by 29

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

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 evidenceassociated 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.

Cited by 20

applications - medical

+ An application of the Dempster–Shafer theory of evidence to the classification of knee function and detection of improvement due to total knee replacement surgery

L Jones, MJ BeynonCA HoltS Roy - Journal of Biomechanics 39, Issue 13, 2006, Pages 2512–2520

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.

Cited by 29


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

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 SilvaA 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.

Cited by 26

applications - environment

Representing uncertainty in silvicultural decisions: an application of the Dempster–Shafer theory of evidence

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

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.

Cited by 28

machine learning - classification – fusion – applications - geoscience

+ Application of Dempster-Shafer evidence theory to unsupervised classification in multisource remote sensing

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.

Cited by 267

machine learning - classification

+ Analysis of evidence-theoretic decision rules for pattern classification

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

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.

Cited by 157


+ 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

Cited by 27

applications - water

+ Interpreting drinking water quality in the distribution system using Dempster–Shafer theory of evidence

R SadiqMJ 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.

Cited by 29

applications - retrieval

+ A Dempster-Shafer indexing for the focussed retrieval of a hierarchically structured document space: Implementation and experiments on a web museum collection

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.

Cited by 47

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.

Cited by 23


+ 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.

Cited by 238

applications - geoscience

+ Multisource data analysis in remote sensing and geographic information systems based on Shafer's theory of evidence

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.

Cited by 19

other theories - endorsements

+ A theory of heuristic reasoning about uncertainty

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

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.

Cited by 95

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

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 isrR, where X is the constrained variable, R is the constraining relation and isr 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

+ A distributed spectrum sensing scheme based on credibility and evidence theory in cognitive radio context

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, and {zengkun, junwang, lsq}

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.

Cited by 137

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 ...

Cited by 173


+ On considerations of credibility of evidence

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

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

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.

Cited by 43

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

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.

Cited by 27

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

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.

Cited by 52

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.

Cited by 26


other theories – expert systems

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

BG BuchananEH Shortliffe – Addison–Wesley, 1984  
Chapter 13 The Dempster-Shafer 
Theory of Evidence 272 Jean Gordon and Edward H. Shortliffe

Cited by 3836


Properties of measures of information in evidence and possibility theories

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

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.

Cited by 26

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

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.

Cited by 16

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.

Cited by 225

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

The following problem from reliability theory is considered. Given a disjunctive normal form (DNF) ϕ = ϕ1 ϕr, we want to find a representation of ϕ into disjoint formulas, i.e. find formulas   such that   and   whenever i ≠ j. In addition, the formulas   must be simple enough that the computation of their probabilities is a simple task. Of course, it is also better if there is only a small number of formulas   in the representation. It has recently been discovered that this problem also appears in the calculation of degrees of support in the context of probabilistic assumption-based reasoning and the Dempster-Shafer theory of evidence (Kohlas and Monney, 1995). In this paper we present a new method to solve this problem, where each formula   is a so-called mix-product. Our method can be applied to any DNF, not only to monotone ones like the method of Heidtmann (1989). However, when applied to monotone formulas, both methods generate the same results. Compared to the algorithm of Abraham (1979) which can also be applied to any DNF, our method is considerably more efficient and will generate a much smaller number of disjoint terms in most cases (see Section 5).

Cited by 38


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


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

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

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.

Cited by 27

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.

Cited by 21



+ Belief functions on real numbers

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

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 R2.

Cited by 128

applications - uncertainty - measurement

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

A FerreroS 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.

Cited by 57


+ 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

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.

Cited by 97

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

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.

Cited by 1292

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.

Cited by 28

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

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.

Cited by 32

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

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.

Cited by 51


+ Independence concepts in evidence theory

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

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.

Cited by 34

debate – uncertainty

+ On quantification of different facets of uncertainty

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

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.

Cited by 38

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

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.

Cited by 22

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].

Cited by 45

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

+ Engine fault diagnosis based on multi-sensor information fusion using Dempster–Shafer evidence theory

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

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 DempsterShafer 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.

Cited by 243

combination - conflict

- Combining belief functions when evidence conflicts

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

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.

Cited by 643

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,

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


Quasi-associative operations in the combination of evidence

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

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.

Cited by 56


A modified combination rule of evidence theory

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

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.

Cited by 42

debate - entropy

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

D DuboisH 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.

Cited by 150

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

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.

Cited by 42

applications - medical

+ Classification of osteoarthritic and normal knee function using three-dimensional motion analysis and the Dempster-Shafer theory of evidence

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.

Cited by 23

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

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.

Cited by 62

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


+ 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.

Cited by 55

machine learning - regression

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

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

This paper introduces a new approach to regression analysis based on a fuzzy extension of belief function theory. For a given input vector  , the method provides a prediction regarding the value of the output variable 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   are considered as sources of partial information on the response variable; the pieces of evidence are discounted as a function of their distance to  , and pooled using Dempster’s rule of combination. The method can cope with heterogeneous training data, including numbers, intervals, fuzzy numbers, and, more generally, fuzzy belief assignments, a convenient formalism for modeling unreliable and imprecise information provided by experts or multi-sensor systems. The performances of the method are compared to those of standard regression techniques using several simulated data sets.

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

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

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.

Cited by 18

debate – combination - evidence

+ The combination of evidence

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

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.

Cited by 104

applications – information systems - risk

+ An information systems security risk assessment model under the Dempster-Shafer theory of belief functions

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

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.

Cited by 137

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.

Cited by 63

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 -

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.

Cited by 210

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

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.

Cited by 109

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

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


Hedging in the combination of evidence

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

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.

Cited by 57

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

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

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.

Cited by 43

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.

Cited by 16

+ 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

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.

Cited by 30

applications – reliability - framework

+ Fault diagnosis of machines based on D–S evidence theory. Part 1: D–S evidence theory and its improvement

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

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

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.

Cited by 16

combination - expert

+ [book] Aggregation and fusion of imperfect information

B Bouchon (editor) – 1998

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.

Cited by 97


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

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.

Cited by 123

applications – medical – classification - fusion

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

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

DOI: 10.1109/TENCON.2006.343718

{Mansoor.Raza, Iqbal. Gondal, David. Green@,

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.

Cited by 14

combination - survey

On the combination of evidence in various mathematical frameworks

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

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.

Cited by 106

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.

Cited by 24

other theories – rough sets

+ 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 1517

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

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.

Cited by 152

applications - risk

+ The words of risk analysis

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

… 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

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.

Cited by 24

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


+ The Dempster–Shafer calculus for statisticians

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

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.

Cited by 134

applications - image

Application of the mathematical theory of evidence to the image cueing and image segmentation problem

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

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.

Cited by 12

combination – other theories - fuzzy

- Fuzzy set connectives as combinations of belief structures

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

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

+ Land use and land cover change prediction with the theory of evidence: a case study in an intensive agricultural region of France

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.

Cited by 15

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

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.

Cited by 183

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

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.

Cited by 72

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

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.

Cited by 12

debate - foundations

+ Understanding evidential reasoning

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

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

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.

Cited by 228

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

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.

Cited by 17

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

no abstract

Cited by 17

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

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

Cited by 19

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.

Cited by 28

decision - utility

Linear utility theory for belief functions

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

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).

Cited by 231



On the unicity of Dempster rule of combination

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

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.

Cited by 165

applications – data mining

+ On subjective measures of interestingness in knowledge discovery

A SilberschatzA 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.

Cited by 428


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

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.

Cited by 19


applications - manufactoring

+ Knitted fabric defect classification for uncertain labels based on Dempster–Shafer theory of evidence

M TabassianR GhaderiR Ebrahimpour - Expert Systems with Applications, Volume 38, Issue 5, May 2011, Pages 5259–5267

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 DempsterShafer 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.

Cited by 16

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

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.

Cited by 72

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.

Cited by 33


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.

Cited by 64

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


+ 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.

Cited by 16

probability transformation

+ On the plausibility transformation method for translating belief function models to probability models

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

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

+ Some classification algorithms integrating Dempster-Shafer theory of evidence with the rank nearest neighbor rules

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.

Cited by 15


+ The consensus operator for combining beliefs

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

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.

Cited by 171

frameworks – generalized evidence theory

The interpretation of generalized evidence theory

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

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.

Cited by 20

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.

Cited by 222

debate - foundations

+ In Defense of Probability

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.

Cited by 397



+ 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.

Cited by 22


Mathematical foundations of evidence theory

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

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.

Cited by 32

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.

Cited by 55

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.

Cited by 17

applications - engineering

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

W Feilhauer, E Handschin – Proc. of the 9th 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.

Cited by 12

applications - engineering

A design optimization method using evidence theory

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

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.

Cited by 112

other theories - fuzzy

Generalized probabilities of fuzzy events from fuzzy belief structures

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

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.

Cited by 141


applications - communications

+ An evidential model of distributed reputation management

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

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.

Cited by 615


fusion – image

Introduction of neighborhood information in evidence theory and application to data fusion of radar and optical images with partial cloud cover

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

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.

Cited by 87


other theories – possibility - language

- On modeling of linguistic information using random sets

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

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.

Cited by 64


other theories – incidence - comparison

+ A comprehensive comparison between generalized incidence calculus and the Dempster-Shafer theory of evidence

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

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.

Cited by 13


machine learning - relaxation

Evidence-based pattern-matching relaxation

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

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.

Cited by 14


multicriteria decision

+ Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: the application of evidence theory

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

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.

Cited by 19


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

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.

Cited by 67


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

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.

Cited by 18


foundations - uncertainty

+ Variants of uncertainty

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

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.

Cited by 518

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

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.

Cited by 46


computation - sampling

+ A sampling-based computational strategy for the representation of epistemic uncertainty in model predictions with evidence theory

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

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 naïve sampling-based (i.e., unsophisticated Monte Carlo) procedures are impracticable due to computational cost.

Cited by 88


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).

Cited by 435

applications - engineering

+ Application of DempsterShafer theory in condition monitoring applications: A case study

Parikh, CR, Pont, MJ, and Jones - Pattern Recognition Letters 22, 777–785, 2001

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

+ Using the Dempster–Shafer theory of evidence with a revised lattice structure for activity recognition

J Liao, Y Bi, C Nugent – IEEE Transactions on Information Technology in Biomedicine 15, No 1, 74-82, 2011

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%.

Cited by 18


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.

Cited by 34


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

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.

Cited by 11


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.

Cited by 117

machine learning – decision trees

Multiple binary decision tree classifiers

S Shlien - Pattern Recognition 23, Issue 7, 1990, Pages 757–763

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.

Cited by 73


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.

Cited by 27


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

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.

cited by 9


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)

Cited by 12

other theories

+ A mathematical theory of evidence for GLS Shackle

G Fioretti - Mind & Society 2, Issue 1, pp 77-98, 2001

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.

Cited by 14


[book] Advances in Dempster-Shafer theory of evidence

J Kacprzyk, M Fedrizzi (editors) - Wiley, New York, 1994

Cited by 14

applications – activity recognition

- Activity recognition for Smart Homes using Dempster-Shafer theory of Evidence based on a revised lattice structure

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%.

Cited by 10


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

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


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

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.

Cited by 12

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

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.

Cited by 149

propagation – other theories - comparison

+ Equivalence of methods for uncertainty propagation of real-valued random variables

HM ReganS FersonD Berleant - International Journal of Approximate Reasoning 36, Issue 1, April 2004, Pages 1–30

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.

Cited by 103

other theories - capacities

Decomposable capacities, distorted probabilities and concave capacities

A Chateauneuf - Mathematical Social Sciences 31, Issue 1, February 1996, Pages 19–37

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.

Cited by 53

applications - agents

+ Situation awareness in intelligent agents: Foundations for a theory of proactive agent behavior

R SoL 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.

Cited by 45

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

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

+ Application of Dempster–Shafer theory in fault diagnosis of induction motors using vibration and current signals

BS Yang, KJ Kim - Mechanical Systems and Signal Processing 20, Issue 2, February 2006, Pages 403–420

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.

Cited by 141


applications – retrieval - empirical

+ Experimenting on Dempster-Shafer's theory of evidence in information retrieval

I RuthvenM Lalmas - Journal of Intelligent Information Systems, 1997


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.

Cited by 9


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

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.

Cited by 11


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.

Cited by 32



+ Attribute reduction based on evidence theory in incomplete decision systems

W-Z Wu - Information Sciences 178, Issue 5, 1 March 2008, Pages 1355–1371

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.

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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