Some Decision Theory for Sets of Probabilities

• Activity Number: 296
• Type: Invited
• Date/Time: Wednesday, August 14, 2002 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Stat. Sciences*
• Author(s): Teddy Seidenfeld*+
• Affiliation(s): Carnegie Mellon University
• Keywords: Robust Bayes analysis ; Ordering postulate ; Independence postulate ; Gamma-minimax

Abstract

Appeal to sets of probability distributions, rather than a single distribution, has proven useful in Robust Bayesian inference, modeling consensus, and in the analysis of such recalcitrant decision problems as the so-called "paradoxes" of Allais and Ellsberg. This paper offers a review and comparison of several approaches to making decisions using sets of probabilities. From the perspective of axiomatic Subjective Expected Utility Theory, alternative decision rules are evaluated for violating either: 1.) the ordering; or 2.) independence postulates. In that regard, the consequences for statistical sequential decision-making using these rules include, respectively: 1.) the anomaly of negative utility for free information; and 2.) various known phenomena--e.g., sensitivity to the stopping rule, associated with failures of the likelihood principle. Decision rules to be discussed include the Gamma-minimax rule, rules based on a JND interpretation of sets of probabilities, and lexicographic rules that require that admissible options maximize expected utility with respect to some probability distribution in the set.