Conditional Events and Probability in the Approach to Artificial Intelligence through Coherence

Abstract

By relying on the most general concept of event as a proposition, our approach refers to an arbitrary family of conditional events, which represent uncertain statements in an expert system. Then probability is interpreted as a measure of the degree of belief in a proposition in a given context, which is expressed in turn by another proposition. Usually it is not realistic to make probability evaluations for all possible envisaged conditional events: yet, requiring coherence of a function P, assessed only for a few conditional events of initial interest, entails that this P is a conditional probability. So the assignment of P can be based on the check of coherence, which amounts to the study of the compatibility of some linear systems, whose unknowns are the probabilities of the atoms generated by the given events. It is then possible to extend probability assessments, preserving coherence, by a step-by-step assignment to further events, leading in general to not necessarily unique values of the ‘new’ probabilities, possibly belonging to suitable closed intervals. Dealing with coherence and the relevant concepts of conditional events and probability is less trivial than it may appear and gives rise to some delicate and subtle problems.