Imprecise Probabilistic Beliefs

Imprecise probabilistic beliefs are modelled as incomplete comparative likelihood relations admitting a multiple-prior representation. We provide an axiomatization of such relations for the case in which the set of priors is “convex-ranged”. We also show that the multiple-prior representation is unique whenever the set of priors is “almost-convex-ranged”. Such uniqueness ensures the adequacy of likelihood relations as models of probabilistic beliefs. In the second part of the paper, two conditions relating preferences and specified probabilistic beliefs are proposed. The weaker one requires simply that the decision maker prefer betting on events in line with specified likelihood comparisons. A somewhat stronger one amounts to a requirement of probabilistic sophistication relative to the specified probabilistic beliefs; it leads to an AnscombeAumann like structure derived in an epistemically enriched Savage framework. ∗e-mail: [email protected] ; homepage: http://www.econ.ucdavis.edu/faculty/nehring/ †Precursors of some of the ideas of this paper were presented in December 1996 at LOFT2 in Torino, Italy, July 2001 at RUD in Venice, Italy, and at the University of Heidelberg, and June 2002 at RUD in Paris.