An Interpretation of Consistent Belief Functions in Terms of Simplicial Complexes

In this paper we pose the study of consistent belief functions (cs.b.f.s) in the framework of the geometric approach to the theory of evidence. As cs.b.f.s are those belief functions whose plausibility assignment is a possibility distribution, their study is a step towards a unified geometric picture of a wider class of fuzzy measures. We prove that, analogously to consonant belief functions, cs.b.f.s form a simplicial complex, and point out the similarity between the consistent complex and the complex of singular belief functions, i.e. belief functions whose core is a proper subset of their domain. Finally, we argue that the notion of complex brings together the possibilistic and probabilistic approximation problems by introducing a convex decomposition of b.f.s in terms of “consistent coordinates” on the complex, closely related to the pignistic transformation.