On transformations of belief functions to probabilities

2014

Ann. Comb.

The study of finite non-additive measures or "belief functions" has been recently posed in connection with combinatorics and convex geometry. As a matter of fact, as belief functions are completely specified by the associated belief values on the events of the frame on which they are defined, they can be represented as points of a Cartesian space. The space of all belief functions B or "belief space" is a simplex whose vertices are BF focused on single events. In this paper we present an alternative description of the space of belief functions in terms of differential geometric notions. The belief space possesses indeed a recursive bundle structure inherently related to the mass assignment mechanism, in which basic probability is recursively assigned to events of increasing size. A formal proof of the decomposition of B together with a characterization of bases and fibers as simplices are provided.

Section: Introductionmentioning

confidence: 99%

On the Fiber Bundle Structure of the Space of Belief Functions

2014

Ann. Comb.

“…Nevertheless, the relation between belief and probability in the theory of evidence has been an important subject of study, and a number of papers have been published on the issue of probability transform [13]. A decision based approach to the problem is the foundation of Smets' "Transferable Belief Model" [25], in which belief functions are defined directly in terms of basis belief assignments ("credal" level), while decisions are made via the pignistic probability BetP [b]…”

Section: Introductionmentioning

confidence: 99%

“…The notion of relative belief transform (under the name of "normalized belief of singletons") has first been proposed by Daniel [13]. Some preliminary analyses of the relative belief transform and its close relationship with the (relative) plausibility transform have been presented in [9,10].…”

Section: Introductionmentioning

confidence: 99%

Game-Theoretical Semantics of Epistemic Probability Transformations

Advances in Intelligent and Soft Computing

2012

Probability transformation of belief functions can be classified into different families, according to the operator they commute with. In particular, as they commute with Dempster's rule, relative plausibility and belief transforms form one such "epistemic" family, and possess natural rationales within Shafer's formulation of the theory of evidence, while they are not consistent with the credal or probability-bound semantic of belief functions. We prove here, however, that these transforms can be given in this latter case an interesting rationale in terms of optimal strategies in a non-cooperative game.

“…As probability measures or Bayesian belief functions are just a special class of b.f.s (for which m(A) = 0 when |A| > 1), the relation between beliefs and probabilities plays a major role in the theory of evidence [9,14,23,11,12,13,2]. Tessem [21], for instance, incorporated only the highest-valued focal elements in his m klx approximation; a similar approach inspired the summarization technique formulated by Lowrance et al [15].…”

Section: Previous Work On Bayesian Approximationmentioning

confidence: 99%

“…In the case ofpl b , even though the latter does not commute with affine combination (the relation being somehow more complex [5]) we can still prove that it commutes with convex closure (9). Using this tools we can find the region of the probability simplex P spanned by the Bayesian transformation of a certain convex region Cl(b 1 , ..., b k ) of b.f.s.…”

Section: K)mentioning

confidence: 99%

Semantics of the Relative Belief of Singletons

Advances in Soft Computing