Showing why Measures of Quantified Beliefs are Belief Functions

Abstract

This chapter highlights the fact that quantified beliefs should be represented by belief functions. The purpose of any mathematical model is for representing quantified beliefs—weighted opinions that can be supported either by defending convincing definitions with illustrative examples or by producing a set of axioms that justify it. The mathematical function that can represent quantified beliefs should be a Choquet capacity monotone of order 2. In order to show that it must be monotone of order infinite, thus a belief function, several extra rationality requirements are proposed the chapter. One of them is based on the negation of a belief function, a concept introduced by Dubois and Prade. In any model for quantified beliefs, an agent, the belief holder, and a finite frame of discernment, denoted by Ω, are considered. If revision of any measure representing quantified beliefs can be represented by a matrix multiplication of the initial beliefs, then the Mobius mass related to the measure must be non negative. This implies that any measure representing quantified beliefs is a belief function.