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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.

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In this paper, drawing inspiration from the commutativity results which hold for a number of Bayesian approximations of belief functions (like pignistic function and relative plausibility of singletons) we study the properties of a new probabilistic approximation of belief functions derived from geometric methods: the intersection probability. The intersection probability inherits its name from the fact that, when combined with a Bayesian function through Dempster’s rule, it is equivalent to the intersection of the line joining a pair of belief and plausibility functions with the affine space of Bayesian pseudo belief functions. Its relation with the convex closure operator in the Cartesian space is analyzed, and equivalent conditions under which they commute are given, showing its similarity with orthogonal projection and pignistic transformation. A thorough analysis of the distance between intersection probability and pignistic function in a case study is conducted, and stringent equivalence relations in terms of mass equi-distribution inferred from it.

In this paper 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. We compute their Moebius inverses, which we call basic plausibility and commonality assignments. As these results are achieved in the framework of the geometric approach to uncertainty measures, the equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. We can then describe the point-wise geometry of these sum functions in terms of rigid transformations mapping them onto each other. Combination rules can be applied to plausibility and commonality functions through their Moebius inverses, leading to interesting applications of such inverses to the probabilistic transformation problem.

In this paper we discuss the semantics and properties of the relative belief transform, a probability transformation of belief functions closely related to the classical plausibility transform. We discuss its rationale in both the probability-bound and Shafer’s interpretations of belief functions. Even though the resulting probability (as it is the case for the plausibility transform) is not consistent with the original belief function, an interesting rationale in terms of optimal strategies in a non-cooperative game can be given in the probability-bound interpretation to both relative belief and plausibility of singletons. On the other hand, we prove that relative belief commutes with Dempster’s orthogonal sum, meets a number of properties which are the duals of those met by the relative plausibility of singletons, and commutes with convex closure in a similar way to Dempster’s rule. This supports the argument that relative plausibility and belief transform are indeed naturally associated with the D-S framework, and highlights a classification of probability transformations in two families, according to the operator they relate to. Finally, we point out that relative belief is only a member of a class of “relative mass” mappings, which can be interpreted as low-cost proxies for both plausibility and pignistic transforms.

probability intervals

In this work we extend the geometric approach to the theory of evidence in order to study the geometric behavior of the two quantities inherently associated with a belief function. i.e. the plausibility and commonality functions. After introducing the analogous of the basic probability assignment for plausibilities and commonalities, we exploit it to understand the simplicial form of both plausibility and commonality spaces. Given the intuition provided by the binary case we prove the congruence of belief, plausibility, and commonality spaces for both standard and unnormalized belief functions, and describe the point-wise geometry of these sum functions in terms of the rigid transformation mapping them onto each other. This leads us to conjecture that the D-S formalism may be in fact a geometric calculus in the line of geometric probability, and opens the way to a wider application of discrete mathematics to subjective probability.