$L_{p}$ Consonant Approximations of Belief Functions

Abstract

In this paper we solve the problem of approximating a belief measure with a necessity measure or "consonant belief function" in a geometric framework. Consonant belief functions form a simplicial complex in both the space of all belief functions and the space of all mass vectors: partial approximations are first sought in each component of the complex, while global solutions are selected among them. As a first step in this line of study, we seek here approximations which minimize L p norms. Approximations in the mass space can be interpreted in terms of mass redistribution, while approximations in the belief space generalize the maximal outer consonant approximation. We compare them with each other and with other classical approximations, and illustrate them with the help of a running example. Index TermsTheory of evidence, possibility theory, consonant belief functions, geometric approach, simplicial complex, (outer) consonant approximation, isopignistic function, L p norms.

New Distance Measures of Evidence Based on Belief Intervals

“…A distance or dissimilarity measure of evidence [6] can describe the degree of dissimilarity or similarity between bodies of evidence (BOEs), which has attracted more and more research interest recently and has been widely used in applications such as algorithm evaluation [7,8] or optimization, clustering analysis, etc. Among the different measures proposed in the literature, Jousselme's distance of evidence [9] and Tessem's distance [10] (also called the betting commitment distance or the pignistic probability distance) are most frequently used.…”

The intersection probability: betting with probability intervals

“…One can distinguish between an affine family of probability transformations [20] (those which commute with affine combination in the belief space), and an epistemic family of transforms [19], formed by the relative belief and relative plausibility of singletons [28,27,37,46,33], which possess dual properties with respect to Dempster's sum [24]. The problem of finding the possibility measure which best approximates a given belief function [2] can also be approached in geometric terms [29,47,39,40]. In particular, approximations induced by classical Minkowski norms can be derived and compared with classical outer consonant approximations [72].…”

A Distance Measure of Interval-valued Belief Structures

“…Among the tools developed to work with DST, distances have recently received increased attention and much work on measuring the distance or dissimilarity between two belief functions has emerged (Jousselme & Maupin 2012). The distance measure can describe the degree of dissimilarity or similarity between bodies of evidence (BOE), and that has been proposed as a tool in various applications including conflict evidence combination (Deng et al 2004;Martin et al 2008), clustering analysis (Deneux 2000), learning models (Zouhal 1998) and belief function approximation (Cuzzolin 2011;Klein et al 2016;Tessem 1993).…”

“…Within the framework of the transferable belief model (TBM), Denoeux extended the main concepts of DST, which include credibility, plausibility, combination and normalization which lay the theoretical foundations of IBS. Most research fields of IBS involve combination rule (Fu & Yang 2012, 2011Sevastianov 2012;Song et al 2014;Wang 2007), normalization (Sevastjanov et al 2010;Xu et al 2012), and uncertainty measure (Jiang 2017;Son 2016). However, few are concerned with the distance within the framework of IBS.…”