Separable qualitative capacities

Abstract

The aim of this paper is to define the counterpart of separable belief functions for capacities valued on a finite totally ordered set. Evidence theory deals with the issue of merging unreliable elementary testimonies. Separable belief functions are the results of this merging process. However not all belief functions are separable. Here, we start with a possibility distribution on the power set of a frame of discernment that plays the role of a basic probability assignment. It turns out that any capacity can be induced by the qualitative counterpart of the definition of a belief function (replacing sum by max). Then, we consider a qualitative counterpart of Dempster rule of combination applied to qualitative capacities, via their qualitative Möbius transforms. We study the class of capacities, called separable capacities, that can be generated by applying this combination rule to simple support capacities, each focusing on a subset of the frame of discernment. We compare this decomposition with the one of general capacities as a maximum over a finite set of necessity measures. The relevance of this framework to the problem of information fusion from unreliable sources is explained.