Decision-Making with Belief Functions: a Review

Approaches to decision-making under uncertainty in the belief function framework are reviewed. Most methods are shown to blend criteria for decision under ignorance with the maximum expected utility principle of Bayesian decision theory. A distinction is made between methods that construct a complete preference relation among acts, and those that allow incomparability of some acts due to lack of information. Methods developed in the imprecise probability framework are applicable in the Dempster-Shafer context and are also reviewed. Shafer's constructive decision theory, which substitutes the notion of goal for that of utility, is described and contrasted with other approaches. The paper ends by pointing out the need to carry out deeper investigation of fundamental issues related to decision-making with belief functions and to assess the descriptive, normative and prescriptive values of the different approaches

Making use of partial knowledge about hidden states in HMMs : an approach based on belief functions.

International audienceThis paper addresses the problem of parameter estimation and state prediction in Hidden Markov Models (HMMs) based on observed outputs and partial knowledge of hidden states expressed in the belief function framework. The usual HMM model is recovered when the belief functions are vacuous. Parameters are learnt using the Evidential Expectation- Maximization algorithm, a recently introduced variant of the Expectation-Maximization algorithm for maximum likelihood estimation based on uncertain data. The inference problem, i.e., finding the most probable sequence of states based on observed outputs and partial knowledge of states, is also addressed. Experimental results demonstrate that partial information about hidden states, when available, may substantially improve the estimation and prediction performances

An Axiomatic Utility Theory for Dempster-Shafer Belief Functions

International audienceThe main goal of this paper is to describe an axiomatic utility theory for Dempster-Shafer belief function lotteries. The axiomatic framework used is analogous to von Neumann-Morgenstern's utility theory for proba-bilistic lotteries as described by Luce and Raiffa. Unlike the probabilistic case, our axiomatic framework leads to interval-valued utilities, and therefore, to a partial (incomplete) preference order on the set of all belief function lotteries. If the belief function reference lotteries we use are Bayesian belief functions, then our representation theorem coincides with Jaffray's representation theorem for his linear utility theory for belief functions. We illustrate our framework using some examples discussed in the literature. Finally, we compare our decision theory with those proposed by Jaffray and Smets

BPEC: Belief-Peaks Evidential Clustering

International audienceThis paper introduces a new evidential clustering method based on the notion of "belief peaks" in the framework of belief functions. The basic idea is that all data objects in the neighborhood of each sample provide pieces of evidence that induce belief on the possibility of such sample to become a cluster center. A sample having higher belief than its neighbors and located far away from other local maxima is then characterized as cluster center. Finally, a credal partition is created by minimizing an objective function with the fixed cluster centers. An adaptive distance metric is used to fit for unknown shapes of data structures. We show that the proposed evidential clustering procedure has very good performance with an ability to reveal the data structure in the form of a credal partition, from which hard, fuzzy, possibilistic and rough partitions can be derived. Simulations on synthetic and real-world datasets validate our conclusions

k-EVCLUS: Clustering Large Dissimilarity Data in the Belief Function Framework

International audienceIn evidential clustering, the membership of objects to clusters is considered to be uncertain and is represented by mass functions, forming a credal partition. The EVCLUS algorithm constructs a credal partition in such a way that larger dissimilarities between objects correspond to higher degrees of conflict between the associated mass functions. In this paper, we propose to replace the gradient-based optimization procedure in the original EVCLUS algorithm by a much faster iterative row-wise quadratic programming method. We also show that EVCLUS can be provided with only a random sample of the dissimilarities, reducing the time and space complexity from quadratic to linear. These improvements make EVCLUS suitable to cluster large dissimilarity datasets

Transformation de scores SVM en fonctions de croyance

National audienceLa combinaison de plusieurs classifieurs, entrainés à partir de données ou caractéristiques distinctes, présente de nombreux intérêts pour les problèmes d'apprentissage supervisé. L'une des difficultés majeures de la combinaison est de représenter les sorties des différents classifieurs sous une forme commune, dans la plupart des cas sous la forme d'une probabilité à postériori. Cette transformation, appelée calibration, joue un rôle central dans la combinaison. Dans cet article, nous étendons les approches classiques de calibration probabilistes en utilisant la théorie des fonctions de croyance. Nous montrons, notamment, l'importance d'utiliser une borne inférieure et supérieure plutôt qu'une simple mesure probabiliste. Des résultats expérimentaux sur la transformation de scores SVM montrent l'apport des fonctions croyances

An evidence-based approach to damage location on an aircraft structure

International audienceThis paper discusses the use of evidence-based classifiers for the identification of damage. In particular, a neural network approach to Dempster-Shafer theory is demonstrated on the damage location problem for an aircraft wing. The results are compared with a probabilistic classifier based on a multi-layer perceptron neural network and shown to give similar results. The question of fusing classifiers is considered and it is shown that a combination of the Dempster-Shafer and MLP classifiers gives a significant improvement over the use of individual classifiers for the aircraft wing data