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Over time, there have hen refinements in the way that probabilitydistributions are used for representing beliefs. Models which rely on singleprobability distributions depict a complete ordering among the propositions ofinterest, yet human beliefs are sometimes not completely ordered. Non-singletonsets of probability distributions can represent partially ordered beliefs.Convex sets are particularly convenient and expressive, but it is known thatthere are reasonable patterns of belief whose faithful representation requireless restrictive sets. The present paper shows that prior ignorance about threeor more exclusive alternatives and the emergence of partially ordered beliefswhen evidence is obtained defy representation by any single set ofdistributions, but yield to a representation baud on several uts. The partialorder is shown to be a partial qualitative probability which shares someintuitively appealing attributes with probability distributions.

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