Consistent transformations of belief functions

TL;DR

This paper explores methods for transforming belief functions into consistent forms using Lp norm minimization, comparing mass-based and belief-based approaches to ensure coherence in evidence reasoning.

Contribution

It introduces a systematic approach to consistent belief function transformations using classical Lp norms in both mass and belief spaces, with detailed comparisons and interpretations.

Findings

Mass and belief space transformations differ in how they reassign belief mass.

Focussed consistent transformations distinguish focal elements based on their focus.

Examples illustrate the practical differences between the approximation methods.

Abstract

Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[.] in a reasoning process to guarantee coherence can therefore be desirable, and generalizes similar techniques in classical logic. Transformations can be obtained by minimizing an appropriate distance measure between the original belief function and the collection of consistent ones. We focus here on the case in which distances are measured using classical Lp norms, in both the "mass space" and the "belief space" representation of belief functions. While mass consistent approximations reassign the mass not focussed on a chosen element of the frame either to the whole frame or to all supersets of the element on an equal basis, approximations in the belief space do distinguish these focal elements according to the "focussed consistent transformation" principle. The different approximations are interpreted and compared, with the help of examples.