Credal semantics of Bayesian transformations

Systems, Man, and Cybernetics, Part B: …, 2010

In this paper we propose a credal representation of the interval probability associated with a belief function (b.f.), and show how it relates to several classical Bayesian transformations of b.f.s through the notion of “focus” of a pair of simplices. While a belief function corresponds to a polytope of probabilities consistent with it, the related interval probability is geometrically represented by a pair of upper and lower 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 of singletons, relative plausibility of singletons and intersection probability can all be described as foci of different pairs of simplices in the region of all probability measures. The formulation of frameworks similar to the Transferable Belief Model for such Bayesian transformations appears then at hand.

IEEE Trans. on Systems, Man, and Cybernetics B ( …, 2009

Logics in Artificial Intelligence, 2008

International Journal of Approximate Reasoning, 2012

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.

Proc. of ISAIM, 2008

2003

Advances in Soft Computing, 2000

Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian states. The pignistic map is not a homomorphism between these same spaces. We demonstrate an impact this may have on robust decision making for frames of cardinality at least 3. We adapt the measure zero reflection property of some maps between probability spaces to define a category of belief states having plausibility zero reflecting functions as morphisms. Our definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map is a functor valued in this category.

The study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form two homogeneous groups, which we call "affine" and "epistemic" families. In this paper we focus on relative plausibility, belief, and uncertainty of probabilities of singletons, the "epistemic" family. They form a coherent collection of probability transformations in terms of their behavior with respect to Dempster's rule of combination. We investigate here their geometry in both the space of all pseudo belief functions and the probability simplex, and compare it with that of the affine family. We provide sufficient conditions under which probabilities of both families coincide.

2018

In uncertainty theories, a common problem is to define how we can extend relations between sets (e.g., inclusion, ranking, consistency, ...) to corresponding notions between uncertainty representations. Such definitions can then be used to perform the same operations as those that are done for sets: measuring information content, ordering alternatives or checking consistency, to name a few. In this paper, we propose a general way to extend set relations to belief functions, using constrained stochastic matrices to identify those belief functions in relation. We then study some properties of our proposal, as well as its relations with existing works focusing on peculiar relations.

2014

Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[•] in a reasoning process to guarantee coherence can therefore be desirable, and generalizes similar techniques in classical logics. Transformations can be obtained by minimizing an appropriate distance measure between the original belief function and the collection of consistent ones. We focus here on the case in which distances are measured using classical L p norms, in both the "mass space" and the "belief space" representation of belief functions. While mass consistent approximations reassign the mass not focussed on a chosen element of the frame either to the whole frame or to all supersets of the element on an equal basis, approximations in the belief space do distinguish these focal elements according to the "focussed consistent transformation" principle. The different approximations are interpreted and compared, with the help of examples.