Schervish: Finitely-Additive Decision Theory

by Robby

UNIVERSITY SEMINAR ON LOGIC, PROBABILITY, AND GAMES
Finitely-Additive Decision Theory
Mark Schervish (Carnegie Mellon)
4:10 pm, Friday, February 16th, 2018
Faculty House, Columbia University

Abstract. We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value ∞. No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we find very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, every decision problem has a minimal complete class and all admissible rules are Bayes rules. Also, every decision problem has a minimax rule and a least-favorable distribution and every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values.  This talk will focus on some examples and the major differences between finitely-additive and countably-additive decision theory.  This is joint work with Teddy Seidenfeld, Jay Kadane, and Rafael Stern.