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18
The geometry of consonant belief functions: simplicial complexes of necessity measures
 Fuzzy Sets and Systems
, 2010
"... In this paper we extend the geometric approach to the theory of evidence in order to include the class of necessity measures, represented on a finite domain of “frame” by consonant belief functions (b.f.s). The correspondence between chains of subsets and convex sets of b.f.s is studied and its prop ..."
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In this paper we extend the geometric approach to the theory of evidence in order to include the class of necessity measures, represented on a finite domain of “frame” by consonant belief functions (b.f.s). The correspondence between chains of subsets and convex sets of b.f.s is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of “simplicial complex”. In particular we focus on the set of outer consonant approximations of a belief function, showing that for each maximal chain of subsets these approximations form a polytope. The maximal such approximation with respect to the weak inclusion relation between b.f.s is one of the vertices of this polytope, and is generated by a permutation of the elements of the frame.
Semantics of the relative belief of singletons
 International Workshop on Uncertainty and Logic UNCLOG’08
, 2008
"... 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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Cited by 12 (9 self)
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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.
On the Properties of the Intersection Probability
, 2007
"... 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 geo ..."
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Cited by 4 (3 self)
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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 equidistribution inferred from it.
On the relative belief transform
, 2011
"... 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 probabilitybound and Shafer’s interpretations of belief functions. E ..."
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Cited by 2 (1 self)
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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 probabilitybound 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 noncooperative game can be given in the probabilitybound 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 DS 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 lowcost proxies for both plausibility and pignistic transforms.
On the properties of relative plausibilities
"... Abstract In this paper we investigate the properties of the relative plausibility function, the probability built by normalizing the plausibilities of singletons associated with a belief function. On one side, we stress how this probability is a perfect representative of the original belief functio ..."
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Abstract In this paper we investigate the properties of the relative plausibility function, the probability built by normalizing the plausibilities of singletons associated with a belief function. On one side, we stress how this probability is a perfect representative of the original belief function when combined with any arbitrary probability through Dempster's rule. This leads to conjecture that this function should also be the solution of the probabilistic approximation problem, formulated naturally in terms of Dempster's rule. On the other side, the geometric properties of relative plausibilities are studied in the context of the geometric approach to the theory of evidence, yielding a description of the representation property which suggests a sketch for the general proof of our conjecture.
Two kadditive generalizations of the pignistic transform for imprecise decision making
"... The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medica ..."
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The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medical diagnosis, the need for an “imprecise ” approach to decision making arises, in which sets of possible outcomes are compared. We propose here a framework for imprecise decision derived from the TBM, in which belief functions are mapped to kadditive belief functions (i.e., belief functions whose focal elements have maximal cardinality equal to k) rather than Bayesian ones. We do so by introducing two alternative generalizations of the pignistic transform to the case of kadditive belief functions. The latter has several interesting properties: depending on which properties are deemed the most important, the two distinct generalizations arise. The proposed generalized transforms are empirically validated by applying them to imprecise decision in concrete pattern recognition problems.
Credal semantics of Bayesian transformations in terms of probability intervals
 Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on
, 2010
"... probability intervals ..."
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