Affiliations: Department of Computing, Oxford Brookes University, Wheatley Campus, Oxford, UK | [x] Japan Advanced Institute of Science and Technology, Ishikawa, Japan | [y] Nanjing University, Nanjing, China | [z] Osaka University and AFOSR/AOARD, Osaka, Japan
Abstract: In this paper we introduce three alternative combinatorial formulations of the theory of evidence (ToE), by proving that both plausibility and commonality functions share the structure of “sum function” with belief functions. We compute their Moebius inverses, which we call basic plausibility and commonality assignments. In the framework of the geometric approach to uncertainty measures the equivalence of the associated formulations of the ToE is mirrored by the geometric congruence of the related simplices. We can therefore describe the point-wise geometry of these sum functions in terms of rigid transformations mapping them onto each other. Combination rules can be applied to plausibility and commonality functions through their Moebius inverses, leading to interesting applications of such inverses to the probabilistic transformation problem.
Keywords: Theory of evidence, combinatorics, sum function, Moebius inverse, basic plausibility and commonality assignment, convex geometry, simplex, congruence
