An Informal Introduction to Quasi-Bayesian Theory (and Lower Probability, Lower Expectations, Choquet Capacities, Robust Bayesian Methods, etc…) for AI

Abstract

This is an attempt to briefly cover essential aspects of Quasi-Bayesian theory and its cousins: lower previsions, lower probability, lower envelopes and Choquet capacities. All these theories deal with sets of probability distributions; they augment/enrich/generalize/improve (pick your word) the infra-structure of usual Bayesian decision theory. Most of the content of this technical report is available in the World-Wide-Web; the current link to this content is located at http://www.cs.cmu.edu/~fcozman/qBayes.html. This technical report provides an official means of referring to the content; I was asked by a number of people to provide it so that the work can be referred to in technical publications. The spirit of this report is informal; the objective is to simplify the presentation where possible even if that means sacrificing some generality or rigor. I concentrate on the axiomatization given by Giron and Rios [11], which they call Quasi-Bayesian theory. This formulation is simple and general; other theories can easily be derived or explained from it. The name also emphasizes the similarities with usual Bayesian theory and the fact that the theory is a theory of decision. The original Quasi-Bayesian theory by Giron and Rios was quite elegant but did not include discussions of conditionalization and independence; they also did not have a clear statement of decision criteria. This work attempts to contribute to the theory by filling these gaps with ideas that have been proposed in a variety of contexts in the last decade. The goal of this work is to present the theory in a unified, informal format so that its scope can be easily appreciated.