Representation and combination of uncertainty with belief functions and possibility measures

TLDR

Shafer’s theory of evidence is gaining acceptance in AI for managing uncertainty, and a central challenge is combining evidence from multiple sources, with probability and possibility theories subsumed within it. This paper investigates the expressive power of belief functions for knowledge representation and tackles the problem of combining uncertain evidence, focusing on Dempster’s rule. The authors examine empirical and axiomatic foundations of belief functions and possibility measures, propose alternative combination rules for nonreliable, nonexhaustive, inconsistent, and dependent sources, and assess their flexibility and numerical stability relative to Dempster’s rule. They conclude that Dempster’s rule is ill‑suited for conflicting information, whereas combination rules derived from fuzzy set and possibility theory are more flexible and numerically stable.

Abstract

The theory of evidence proposed by G. Shafer is gaining more and more acceptance in the field of artificial intelligence, for the purpose of managing uncertainty in knowledge bases. One of the crucial problems is combining uncertain pieces of evidence stemming from several sources, whether rules or physical sensors. This paper examines the framework of belief functions in terms of expressive power for knowledge representation. It is recalled that probability theory and Zadeh's theory of possibility are mathematically encompassed by the theory of evidence, as far as the evaluation of belief is concerned. Empirical and axiomatic foundations of belief functions and possibility measures are investigated. Then the general problem of combining uncertain evidence is addressed, with focus on Dempster rule of combination. It is pointed out that this rule is not very well adapted to the pooling of conflicting information. Alternative rules are proposed to cope with this problem and deal with specific cases such as nonreliable sources, nonexhaustive sources, inconsistent sources, and dependent sources. It is also indicated that combination rules issued from fuzzy set and possibility theory look more flexible than Dempster rule because many variants exist, and their numerical stability seems to be better.