Algorithms for Imprecise Probabilities

Abstract

This paper reviews local computation algorithms to calculate with imprecise probabilities. Most of the existing work on imprecise probabilities has focused on the particular case of interval probabilities [Amarger et al., 1991; de Campos and Huete, 1993; de Campos et al., 1994; Dubois and Prade, 1994; Fagin and Halpern, 1991; Fertig and Breese, 1990; Fertig and Breese, 1993; van der Gagg, 1991; Grosof, 1986; Nilsson, 1986; Paass, 1988; Quinlan, 1983; Tessen, 1989; Tessem, 1992]. However, a more general approach for imprecise probabilities consists in the use of convex sets of probability measures [de Campos and Moral, 1995; Cano et al., 1993a; Dempster, 1967; Kyburg and Pittarelli, 1992; Levi, 1980; Levi, 1985; Moral, 1993; Pittarelli, 1991; Smith, 1961; Snow, 1986; Stirling and Morrel, 1991; Walley, 1991; Walley, 1996; Wilson and Moral, 1994]. The basic idea is that if for a variable we do not have the exact values of probabilities, we may have a convex set of possible probability distributions. From a behavioral point of view the use of convex sets of probabilities was justified by Walley [1991]. According to this author what distinguishes this theory from the Bayesian one is that imprecision in probability and utility models is admitted. Strict Bayesians assume that for each event there is some betting rate you consider fair: you are ready to bet on either side of the bet. This rate determines the exact value of your subjective probability of the event. Convex sets of probabilities arose by assuming that for each event there is a maximum rate at which you are prepared to bet on it (determining its lower probability) and a minimum rate (determining its upper probability).