Bayesian updating rules and AGM belief revision

Abstract

We interpret the problem of updating beliefs as a choice problem (selecting a posterior from a set of admissible posteriors) with a reference point (prior). We use AGM belief revision to define the support of admissible posteriors after arrival of information, which applies also to zero probability events. We study two classes of updating rules for probabilities: 1) “lexicographic” updating rules where posteriors are given by a lexicographic probability system 2) “minimum distance” updating rules which select the posterior closest to the prior by some metric. We show that an updating rule is lexicographic if and only if it is Bayesian, AGM-consistent and satisfies a weak form of path independence. While not all lexicographic updating rules have a minimum distance representation, we study a sub-class of lexicographic rules, which we call “support-dependent” rules, which admit a minimum distance representation. Finally, we apply our approach to the problem of updating preferences.