A Mathematical Formulation of Dempster-Shafer’s Belief Functions

Abstract

This paper introduces a unified mathematical formulation for both Bayesian approach and Dempster-Shafer’s approach in handling uncertainty in artificial intelligence. Each body of uncertain information in this formulation is an information quadruplet, consisting of a code space, a message space, an interpretation function, and an evidence space. Each information quadruplet contains prior information as well as possible new evidence which may appear later. Bayes rule is used to update the prior information. This paper also introduces an idea of independent information and its combination, in which a combination formula is derived. A conventional Bayesian prior probability measure is the prior information of a special information quadruplet; Bayesian conditioning is the combination of special independent information. A Dempster’s belief function is the belief function of a different information quadruplet; the Dempster combination rule is the combination rule of such independent quadruplets. This paper shows that both the conventional Bayesian approach and Dempster-Shafer’s approach originate from the same mathematical theory.

This work was supported in part by the National Science Foundation tinder grant number IRI-8505735 and a summer research grant of Ball State University.