Dual properties of relative belief of singletons

In this paper we prove that a recent Bayesian approximation of belief functions, the relative belief of singletons, meets a number of properties with respect to Dempster’s rule of combination which mirrors those satisfied by the relative plausibility of singletons. In particular, the related operator commutes with Dempster’s sum of plausibility functions, while the relative belief perfectly represents a plausibility function when combined through Dempster’s rule. This suggests a classification of all Bayesian approximations into two families according to the operator they relate to. Introduction: A new Bayesian approximation The theory of evidence (ToE) (Shafer 1976) extends classical probability theory through the notion of belief function (b.f.), a mathematical entity which independently assigns probability values to sets of possibilities rather than single events. A belief function b : 2 → [0, 1] on a finite set (“frame”) Θ has the form b(A) = ∑ B⊆A mb(B) where mb : 2 → [0, 1], is called “basic probability assignment” (b.p.a.), and meets the positivity mb(A) ≥ 0 ∀A ⊆ Θ and normalization ∑ A⊆Θ mb(A) = 1 axioms. Events associated with non-zero basic probabilities are called “focal elements”. A b.p.a. can be uniquely recovered from a belief function through Moebius inversion: