Geometric Foundations for Interval­Based Probabilities

Vu Ha and Peter Haddawy Decision Systems and Artificial Intelligence Lab Dept. of EE&CS University of Wisconsin­Milwaukee Milwaukee, WI 53201

Abstract

The need to reason with imprecise probabil­ ities arises in a wealth of situations rang­ ing from pooling of knowledge from multi­ ple experts to abstraction­based probabilistic planning. Researchers have typically repre­ sented imprecise probabilities using intervals and have developed a wide array of differ­ ent techniques to suit their particular require­ ments. In this paper we provide an analysis of some of the central issues in representing and reasoning with interval probabilities. At the focus of our analysis is the probability cross­product operator and its interval gen­ eralization, the cc­operator. We perform an extensive study of these operators relative to manipulation of sets of probability distribut­ tions. This study provides insight into the sources of the strengths and weaknesses of various approaches to handling probability intervals. We demonstrate the application of our results to the problems of inference in in­ terval Bayesian networks and projection and evaluation of abstract probabilistic plans.