Ignorance and the Expressiveness of Single- and Set-Valued Probability Models of Belief

Over time, there have been refinements in the way that probability distributions are used for representing beliefs. Models which rely on single probability distnlnrti ons depict a complete ordering among the propositions of interest, yet human beliefs are sometimes not completely ordered. Non-singleton sets of probability distributions can represent partially ordered beliefs. Convex sets are particularly convenient and expressive, but it is known that there are reasonable patterns of belief whose faithful representation require less restrictive sets. The present paper shows that prior ignorance about three or more exclusive alternatives and the emergence of partially ordered beliefs when evidence is obtained defy representation by any single set of distributions, but yield to a representation based on several sets. The partial order is shown to be a partial qualitative probability which shares some intuitively appealing attributes with probability distributions.