Belief Functions on MV-Algebras of Fuzzy Sets: An Overview

Abstract

Belief functions are the measure theoretical objects Dempster-Shafer evidence theory is based on. They are in fact totally monotone capacities, and can be regarded as a special class of measures of uncertainty used to model an agent's degrees of belief in the occurrence of a set of events by taking into account different bodies of evidence that support those beliefs. In this chapter we present two main approaches to extending belief functions on Boolean algebras of events to MV-algebras of events, modelled as fuzzy sets, and we discuss several properties of these generalized measures. In particular we deal with the normalization and soft-normalization problems, and on a generalization of Dempster's rule of combination.

Coherence in the aggregate: A betting method for belief functions on many-valued events

“…As pointed out in [17] (see also [13,14]), since ρ 0 does not coincide in general with the zeroconstant function 0, Bel(0) cannot be ensured to equal 0. We call normalised each belief function Bel on [0, 1] W satisfying Bel(0) = 0.…”

Corrigendum and addendum to 'Universal algebra for general aggregation theory'

“…However, finitely-additive probability measures on MV-algebras are known as states and have also been studied by numerous authors, following the seminal paper by Mundici [12]. An overview of related literature can be found in Flaminio, Godo and Kroupa [9]. One may hope that ultimately an algebraic framework for aggregation theory which does encompass probabilistic opinion pooling may be developed, even though the concept of homomorphy on which our approach was based is not suitable for this purpose.…”