Some Static and Dynamic Aspects of Robust Bayesian Theory

Abstract

In this presentation, I discuss two features of robust Bayesian theory that arise, naturally, when considering more than one decision maker:

• (1) On a question of statics — what opportunities are there for Bayesians to engage in cooperative decision making while conforming the group to a (mild) Pareto principle and expected utility theory?

• (2) On a question of dynamics — what happens, particularly in the short run, when a collection of Bayesian opinions are updated using Bayes’ rule, where each opinion is conditioned on shared data?

In connection with the first problem — to allow for a Pareto efficient cooperative group — I argue for a relaxation of the “ordering” postulate in expected utility theory. I outline a theory of partially ordered preferences, developed in collaboration with Jay Kadane and Mark Schervish (1995), that relies on sets of probability / Utility pairs rather than a single such pair for making reobust Bayesian decisions.

In connection with the second problem — Tim Herron, Larry Wasserman, and I report on an anomalous phenomenon. We call it “dilation,” where conditioning on new shared evidence is certain to enlarge the range of opinions about an event of common interest. Dilation stands in contrast to well-known results about the asymptotic merging of Bayesian decision making to collect “cost-free” data prior to making a terminal decision?

The use of a set of probabilities to represent opinion, rather than the use of a single probability to do so, arises in the theory of random sets, e.g., when the random objects are events from a finite powerset. Then, as is well known, the set of probabilities is just that determined by the lower probability of a belief function. Dilation applies to belief functions, too, as I illustrate.