The Consistency Problem in Belief and Probability Assessments
Andrea Capotorti, Barbara Vantaggi · 1996
Among quantitative approaches for handling partial knowledge in Artificial Intelligence (AI). belief and probability functions are the most common. A domain expert can usually give a partial evaluation only on few events (generally without a particular logic frame). on which he has information at the moment. Hence. if we want to see this evaluation as a restriction of a particular kind of function representing uncertainty, we must prove consistency of these values. To do this we need to introduce computable consistency properties (as de Finetti's coherence principle for probability assessments). The main problem is computational complexity. In this paper we show cases with a right balance between dimensional complexity and frame richness. and belief functions. These two approaches present similar difficulties both for checking the consistency of evaluations and for the numerical complexity of the entities involved the computation. The order of complexity of the latter is usually worse than the one of the former. We provide meaningful examples, both with probability and belief assessments, to illustrate this conflict between freedom of evaluation and consistency. In the general case the proof of consistency is carried out by solving a linear system. We have identified particular logical situations where the problem of solving the systems can be skipped by checking further properties involving only the initial values. PROBABILITY ASSESSMENTS 2 INTRODUCTION 1 Let X be a finite set of events that we will denote with Ei i = 1,..., n. The classical probability theory requires to endow X with a particular frame, the algebra A, so that a probability evaluation is a function