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Credal semantics of Bayesian transformations
"... In this paper we propose a credal representation of the set of interval probabilities associated with a belief function, and show how it relates to several classical Bayesian transformations of belief functions through the notion of “focus ” of a pair of simplices. Starting from the interpretation o ..."
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In this paper we propose a credal representation of the set of interval probabilities associated with a belief function, and show how it relates to several classical Bayesian transformations of belief functions through the notion of “focus ” of a pair of simplices. Starting from the interpretation of the pignistic function as the center of mass of the credal set of consistent probabilities, we prove that relative belief and plausibility of singletons and intersection probability can be described as foci of different pairs of simplices in the simplex of all probability measures. Such simplices are associated with the lower and upper probability constraints, respectively. This paves the way for the formulation of frameworks similar to the transferable belief model for lower, upper, and interval constraints.
Geometric conditional belief functions in the belief space
 7TH INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS, INNSBRUCK, AUSTRIA
, 2011
"... In this paper we study the problem of conditioning a belief function (b.f.) b with respect to an event A by geometrically projecting such belief function onto the simplex associated with A in the space of all belief functions. Defining geometric conditional b.f.s by minimizing Lp distances between b ..."
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In this paper we study the problem of conditioning a belief function (b.f.) b with respect to an event A by geometrically projecting such belief function onto the simplex associated with A in the space of all belief functions. Defining geometric conditional b.f.s by minimizing Lp distances between b and the conditioning simplex in such “belief ” space (rather than in the “mass ” space) produces complex results with less natural interpretations in terms of degrees of belief. The question of weather classical approaches, such as Dempster’s conditioning, can be themselves reduced to some form of distance minimization remains open: the generation of families of combination rules generated by (geometrical) conditioning appears to be the natural prosecution of this line of research.
Geometric conditioning in belief calculus
"... Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of uncertainty, but different approaches to conditioning in that framework have been proposed in the past, leaving the matter un ..."
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Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of uncertainty, but different approaches to conditioning in that framework have been proposed in the past, leaving the matter unsettled. We propose here an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. Two different such simplices can be defined, as each belief function can be represented as either the vector of its basic probability values or the vector of its belief values. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. The question of whether classical approaches, such as for instance Dempster’s conditioning, can also be reduced to some form of distance minimization remains open: the study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research.
On the orthogonal projection of a belief function
"... Abstract. In this paper we study a new probability associated with any given belief function b, i.e. the orthogonal projection π[b] of b onto the probability simplex P. We provide an interpretation of π[b] in terms of a redistribution process in which the mass of each focal element is equally distri ..."
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Abstract. In this paper we study a new probability associated with any given belief function b, i.e. the orthogonal projection π[b] of b onto the probability simplex P. We provide an interpretation of π[b] in terms of a redistribution process in which the mass of each focal element is equally distributed among its subsets, establishing an interesting analogy with the pignistic transformation. We prove that orthogonal projection commutes with convex combination just as the pignistic function does, unveiling a decomposition of π[b] as convex combination of basis pignistic functions. Finally we discuss the norm of the difference between orthogonal projection and pignistic function in the case study of a quaternary frame, as a first step towards a more comprehensive picture of their relation. 1
Alternative Formulations of the Theory of Evidence Based on Basic Plausibility and Commonality Assignments
"... Abstract. In this paper we introduce indeed two alternative formulations of the theory of evidence by proving that both plausibility and commonality functions share the same combinatorial structure of sum function of belief functions, and computing their Moebius inverses called basic plausibility an ..."
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Abstract. In this paper we introduce indeed two alternative formulations of the theory of evidence by proving that both plausibility and commonality functions share the same combinatorial structure of sum function of belief functions, and computing their Moebius inverses called basic plausibility and commonality assignments. The equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. Applications to the probabilistic approximation problem are briefly presented. 1
The Intersection Probability and Its Properties
"... Abstract. In this paper we discuss the properties of the intersection probability, a recent Bayesian approximation of belief functions introduced by geometric means. We propose a rationale for this approximation valid for interval probabilities, study its geometry in the probability simplex with res ..."
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Abstract. In this paper we discuss the properties of the intersection probability, a recent Bayesian approximation of belief functions introduced by geometric means. We propose a rationale for this approximation valid for interval probabilities, study its geometry in the probability simplex with respect to the polytope of consistent probabilities, and discuss the way it relates to important operators acting on belief functions. 1