On the Fiber Bundle Structure of the Space of Belief Functions

Abstract

The study of finite non-additive measures or “belief functions” has been recently posed in connection with combinatorics and convex geometry. As a matter of fact, as belief functions are completely specified by the associated belief values on the events of the frame on which they are defined, they can be represented as points of a Cartesian space. The space of all belief functions \({\mathcal{B}}\) or “belief space” is a simplex whose vertices are BF focused on single events. In this paper, we present an alternative description of the space of belief functions in terms of differential geometric notions. The belief space possesses indeed a recursive bundle structure inherently related to the mass assignment mechanism, in which basic probability is recursively assigned to events of increasing size. A formal proof of the decomposition of \({\mathcal{B}}\) together with a characterization of its bases and fibers as simplices are provided.