A Representation of Partially Ordered Preferences
Abstract or Description
This essay considers decision-theoretic foundations for robust
Bayesian statistics. We modify the approach of Ramsey, de Finetti, Savage
and Anscombe and Aumann in giving axioms for a theory of robust
preferences. We establish that preferences which satisfy axioms for robust
preferences can be represented by a set of expected utilities. In the
presence of two axioms relating to state-independent utility, robust preferences are represented by a set of probability/utility pairs, where the
utilities are almost state-independent (in a sense which we make precise).
Our goal is to focus on preference alone and to extract whatever probability
and/or utility information is contained in the preference relation when
that is merely a partial order. This is in contrast with the usual approach
to Bayesian robustness that begins with a class of "priors" or "likelihoods,"
and a single loss function, in order to derive preferences from these