Evidence, knowledge, and belief functions

This article addresses most of the questions raised by Pearl in the 1990 special issue of the International Journal of Approximate Reasoning on belief functions and belief maintenance in artificial intelligence.

Abstract

This article tries to clarify some aspects of the theory of belief functions especially with regard to its relevance as a model for incomplete knowledge. It is pointed out that the mathematical model of belief functions can be useful beyond a theory of evidence, for the purpose of handling imperfect statistical knowledge. Dempster's rule of conditioning is carefully examined and compared to upper and lower conditional probabilities. Although both notions are extensions of conditioning, they cannot serve the same purpose. The notion of focusing, as a change of reference class, is introduced and opposed to updating. Dempster's rule is good for updating, whereas the other form of conditioning expresses a focusing operation. In particular, the concept of focusing models the meaning of uncertain statements in a more natural way than updating. Finally, it is suggested that Dempster's rules of conditioning and combination can be justified by the Bayers rule itself. On the whole this article addresses most of the questions raised by Pearl in the 1990 special issue of the International Journal of Approximate Reasoning on belief functions and belief maintenance in artificial intelligence.