Moebius inverses of plausibility and commonality functions

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. As they are discovered through geometric methods, the latter inherit the same simplicial geometry of belief functions: The equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of those simplices. Applications to the probabilistic approximation problem are briefly presented. Introduction The theory of evidence (ToE) is one of the most popular uncertainty theory (Shafer 1976; Dempster 1968), in which subjective probability is represented by belief function (b.f.) rather than a Bayesian mass distribution, assigning probability values to sets of possibilities rather than single events. Variants or continuous extensions of the ToE in terms of hints (Kohlas 1995) or allocations of probability (Shafer 1979) have since been proposed. From a combinatorial point of view, in their finite incarnation, b.f.s are sum functions, i.e. functions on the power set 2 = {A ⊆ Θ} of a finite domain Θ b(A) = B⊆A mb(B) induced by a basic probability assignment (b.p.a.) mb : 2 → [0, 1] which is combinatorially the Moebius inverse (Aigner 1979) of b. The same evidence associated with a b.f. is carried by the related plausibility (pl.f.) plb(A) = 1− b(A) and commonality Qb(A) = ∑ B⊇A mb(B) (comm.f.) functions, which lack though a similar coherent mathematical characterization. In this paper we introduce indeed two alternative formulations of the theory of evidence by proving that both pl.f.s and comm.f.s share the same combinatorial structure of sum function, and computing their Moebius inverses which is natural to call basic plausibility and commonality assignments. We achieve this by resorting to a recent geometric approach to the theory of evidence (Cuzzolin 2008) in which belief functions are represented by points of a Cartesian space. Besides giving the overall mathematical structure of the theory of evidence a more elegant symmetry, the Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. notions of b.pl.a.s and b.comm.a.s turn out to be useful when solving problems like finding probabilistic approximations (Smets 1988; Voorbraak 1989; Bauer 1997) of belief functions, or computing the canonical decomposition of support functions. Moreover, as they are discovered through geometric methods, basic plausibility and commonality assignments inherit the same simplicial geometry as that of b.f.s. The equivalence of the associated formulations of the ToE is then mirrored by the geometric congruence of those simplices. After recalling the basic notions of the theory of evidence, we review the geometric approach to the ToE, to later introduce the notions of basic plausibility and commonality assignments as Moebius inverses of pl.f.s and comm.f.s respectively. As a consequence the geometric description of uncertainty is extended to plausibility and commonality functions, and the simplicial structure of the related spaces recovered. The equivalence of the alternative formulations of the ToE is reflected by the congruence of the corresponding simplices in the geometric framework. In the last section some applications of b.pl.a.s to the approximation problem are discussed. Belief, plausibility, and commonality functions Even though belief functions can be given several alternative but equivalent definitions in terms of multi-valued mappings, random sets (Nguyen 1978; Hestir, Nguyen, & Rogers 1991), inner measures (Fagin & Halpern 1988), in Shafer’s formulation (Shafer 1976) a central role is played by the notion of ”basic probability assignment”. A basic probability assignment (b.p.a.) over a finite set (frame of discernment (Shafer 1976)) Θ is a function m : 2 → [0, 1] on its power set 2 = {A ⊂ Θ} such that m(∅) = 0, ∑ A⊆Θ m(A) = 1, m(A) ≥ 0 ∀A ⊂ Θ. Subsets of Θ associated with non-zero values of m are called focal elements. The belief function (b.f.) b : 2 → [0, 1] associated with a basic probability assignment mb on Θ is defined as: