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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 approach allows to compute strong inner and outer approximations which can be combined efficiently using the Fast Möbius Transform algorithm.

In this paper the geometric structure of the space S of the belief functions dened over a discrete set (be- lief 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.

In this paper we adopt the geometric approach to the theory of evidence to study the geometric counterparts of the plausibility functions, or upper probabilities.

Summary. In this paper we introduce the relative belief of singletons as a novel Bayesian approximation of a belief function. We discuss its nature in terms of degrees of belief under several different angles, and its applicability to different classes of belief functions.

In this chapter, we propose a new practical codification of the elements of the Venn diagram in order to easily manipulate the focal elements. In order to reduce the complexity, the eventual constraints must be integrated in the codification at the beginning. Hence, we only consider a reduced hyper power set D Θ r that can be 2 Θ or D Θ. We describe all the steps of a general belief function framework. The step of decision is particularly studied, indeed, when we can decide on intersections of the singletons of the discernment space no actual decision functions are easily to use. Hence, two approaches are proposed, an extension of previous one and an approach based on the specificity of the elements on which to decide. The principal goal of this chapter is to provide practical codes of a general belief function framework for the researchers and users needing the belief function theory.

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An evidence theoretic classification method is proposed in this paper. In order to classify a pattern we consider its neighbours, which are taken as parts of a single source of evidence to support the class membership of the pattern. A single mass function or basic belief assignment is then derived, and the belief function and the pignistic ("betting rates") probability function can be calculated. Then the (posterior) conditional pignistic probability function is calculated and used to decide the class label for the pattern.