PASCAL - Pattern Analysis, Statistical Modelling and Computational Learning

Making Decisions Using Sets of Probabilities: Updating, Time Consistency, and Calibration
Peter Grünwald and Joseph Halpern
Journal of Artificial Intelligence Research (JAIR) Volume 42, pp. 393-426, 2011.

Abstract

We consider how an agent should update her beliefs when her beliefs are represented by a set P of probability distributions, given that the agent makes decisions using the minimax criterion, perhaps the best-studied and most commonly-used criterion in the literature. We adopt a game-theoretic framework, where the agent plays against a bookie, who chooses some distribution from P. We consider two reasonable games that differ in what the bookie knows when he makes his choice. Anomalies that have been observed before, like time inconsistency, can be understood as arising because different games are being played, against bookies with different information. We characterize the important special cases in which the optimal decision rules according to the minimax criterion amount to either conditioning or simply ignoring the information. Finally, we consider the relationship between updating and calibration when uncertainty is described by sets of probabilities. Our results emphasize the key role of the rectangularity condition of Epstein and Schneider.

EPrint Type:Article
Project Keyword:Project Keyword UNSPECIFIED
Subjects:Computational, Information-Theoretic Learning with Statistics
Theory & Algorithms
ID Code:8845
Deposited By:Peter Grünwald
Deposited On:21 February 2012