Maximum Uncertainty Procedures for Interval-Valued Probability Distributions

An interval-valued distribution is a fmite sequence of intervals ([Pl1 ,Pu1 ], ... ,[Pln,Pun]) n n such that O$Plj$Put�l, E Puj�l, and E Plj$1. Interval distributions generalize realj=l j=l valued probability distributions and arise naturally in many situations. They may represent collections of confidence intervals derived from frequency data, imprecisely stated subjective probabilities, known linear equality or inequality constraints, etc. Thus, interval distributions sometimes provide a more realistic characterization of uncertainty than do real-valued probability distributions. In addition, certain practical problems, e.g., inconsistency of overlapping probability estimates over structured spaces, are much less severe for such distributions [Pittarelli, 1989] At the same time, interval distributions can be put to many of the same uses as real-valued distributions. For example, criteria have been developed for decision analysis using probability intervals [Seidenfeld, 1983; Loui et al., 1986; Pittarelli, 1989]. To illustrate a situation in which available information may be summarized by an interval-valued distribution, consider the probabilistic database [Cavallo and Pittarelli, 1987; Pittarelli, 1989] (collection of marginal distributions) D = {PI, P2 }: