Research Theme: Uncertainty Theory
Live Projects
Geometric Approach to Uncertainty
Total Belief Theorem
Statistical Inference with Belief Likelihood
Generalised Logistic Regression

Uncertainty measures can be represented as points of a convex space, and there analysed. In particular, both the space of belief functions and the behaviour of Dempster's combination rule can be described in geometric terms.

The law of total probability can be generalised to the case of belief measures. We derive the analytical form of the simplest solution, and study the structure of the set of all solutions.

We are devising new approach to statistical inference with belief functions based on the notion of belief likelihood, as a direct generalisation of the traditional likelihood function.

Based on the notion of belief likelihood function, the logistic regression framework can indeed be generalised to the case of belief functions, which themselves generalise classical discrete probability measures.

Generalised Max-Entropy Classification
Geometry of General Combination

We are working on a generalised maximum-entropy classification framework, in which the empirical expectation of the feature functions is bounded by the lower and upper expectations associated with the lower and upper probabilities associated with a belief measure.

We build on previous work on the geometry of Dempster's rule to investigate the geometric behaviour of various other combination rules, including Yager's, Dubois', and disjunctive combina- tion, starting from the case of binary frames of discernment. Believabil- ity measures for unnormalised belief functions are also considered. A research programme to complete this analysis is outlined.

Past Projects
Belief and Probability
Belief and Possibility
Conditioning
Combinatorics

Belief functions can be mapped to probability measures to reduce complexity, or for decision making purposes. We distinguish two main families of probability transform: the affine and the epistemic family.

Consonant and consistent belief functions are special classes of BFs which correspond to possibility measures and to consistent knowledge bases, respectively. We analysed their geometry and relation with general belief functions.

Conditioning can be defined for belief functions in several alternative ways. We analysed a geometric notion of conditioning in which the conditional belief function minimises an appropriate distance from the original BF.

We introduce three alternative combinatorial formulations of the theory of evidence (ToE), by proving that both plausibility and commonality functions share the structure of 'sum function' with belief functions.

Betting with Probability Intervals
Algebra and Independence of Frames
Decision Making
Imprecise Hidden Markov Models

Probability intervals [Moral] are an attractive tool for reasoning under uncertainty. We propose the use of the intersection probability, originally deribed for belief functions in our geometric framework, as the most natural probability transform for them.

Families of compatible frames onto which belief functions are defined can be described algebraically as lower and upper semimodular lattices. The notion of independence of sources can also be studied in algebraic terms.

Decision making under uncertainty is an important application of belief theory. We illustrate a number of approaches which make use of probability transforms.

A novel technique to classify time series with imprecise hidden Markov models is presented. The learning of these models is achieved by coupling the EM algorithm with the imprecise Dirichlet model. In the stationarity limit, each model corresponds to an imprecise mixture of Gaussian densities.