Ignorance and the Expressiveness of Single- and Set-Valued Probability
  Models of Belief
  Over time, there have hen refinements in the way that probability
distributions are used for representing beliefs. Models which rely on single
probability distributions 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 baud on several uts. The partial
order is shown to be a partial qualitative probability which shares some
intuitively appealing attributes with probability distributions.
