A Geometric Approach to the Theory of Evidence

Abstract

In this paper we propose a geometric approach to the theory of evidence based on convex geometric interpretations of its two key notions of belief function and Dempster's sum. On one side, we analyze the geometry of belief functions as points of a polytope in the Cartesian space called belief space, and discuss the intimate relationship between basic probability assignment and convex combination. On the other side, we study the global geometry of Dempster's rule by describing its action on those convex combinations. By proving that Dempster's sum and convex closure commute we are able to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function b. Natural applications of these geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined. Index TermsTheory of evidence, belief function, belief space, simplex, Dempster's rule, conditional subspace.

Classification of EEG signals using Dempster Shafer theory and a k-nearest neighbor classifier

“…This theory can be considered as a generalization of the probability theory [5], [6]. In the following a very brief introduction to the basic notions of the theory of evidence is given.…”

On the Orthogonal Projection of a Belief Function

“…Motivated by the search for a meaningful probabilistic approximation of belief functions we introduced the notion of belief space [19], as the space of all belief functions defined on a given frame of discernment Θ. = 2 |Θ| , and can then be represented as a point of R N −1 .…”

“…We will assume the domain Θ fixed, and denote the belief space with B. It is not difficult to prove [19] …”

“…In robust Bayesian statistics, there is a large literature on the study of convex sets of probability distributions [14][15][16][17]. On our side, in a series of works [18,19] we proposed a geometric interpretation of the theory of evidence in which belief functions are represented as points of a simplex called belief space B, a polytope whose vertices are all the b.f.s focused on a single event A, m b (A) = 1, m b (B) = 0 ∀B = A. The region P of Bayesian b.f.s is also a simplex, part of the border of B.…”

Game-Theoretical Semantics of Epistemic Probability Transformations

“…Other proposals have been recently brought forward by Dezert et al [16], Burger [3], Sudano [27] and others, based on redistribution processes similar to that of the pignistic transform. In addition, two new Bayesian approximations of belief functions have been derived from purely geometric considerations [7] in the context of the geometric approach to the ToE [8], in which belief and probability measures are represented as points of a Cartesian space.…”