Toward a general theory of conditional beliefs

Abstract

We consider a class of general decomposable measures of uncertainty, which encompasses (as its most specific elements, with respect to the properties of the rules of composition) probabilities, and (as its most general elements) belief functions. The aim, using this general context, is to introduce (in a direct way) the concept of conditional belief function as a conditional generalized decomposable measure φ(·|·), defined on a set of conditional events. Our main tool will be the following result, that we prove in the first part of the article and which is a sort of converse of a well-known result (i.e., a belief function is a lower probability): a coherent conditional lower probability P(·|K) extending a coherent probability P(Hi)—where the events His are a partition of the certain event Ω and K is the union of some (possibly all) of them—is a belief function. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 229–259, 2006.