A coherent qualitative Bayes´ theorem and its application in artificial intelligence

Qualitative probability deals with properties of a binary relation to be read `is no more probable than´, or its strict companion `is less probable than´, on a set of conditional events, such as `A given H´. From the point of view of applications, it is clearly very significant not assuming any specific structure for the set on which the qualitative probability is defined. This situation is typical for the conditional events representing uncertain statements in artificial intelligence, for example in expert systems. The intuitive meaning of the binary relation is simply `A given H is no more probable than B given K´, i.e., none of the usual axioms is a priori adopted. The authors introduce instead, for this binary relation, coherence conditions which are related to de Finetti´s coherent systems of bets and study their implications with reference to the properties of qualitative probability and to the possibility of its numerical representation. A qualitative Bayes´ theorem is then derived. It allows the comparison of the conditional events `H given E´ and `K given E´ once the ordering between `E given H´ and `E given K´ and that between H and K are both given. Some applications are considered