Relative Maximum Likelihood Updating of Ambiguous Beliefs

TL;DR

This paper introduces the Relative Maximum Likelihood (RML) updating rule for ambiguous beliefs, unifying and extending existing methods like Bayesian and Maximum Likelihood updates, with axiomatic foundations and new characterizations.

Contribution

It proposes RML as a new updating rule for ambiguous beliefs, providing axiomatic characterization and including known rules as special cases.

Findings

RML encompasses Bayesian and ML as special cases.

A new axiom characterizes ML updating.

Addresses a long-standing open question in the literature.

Abstract

This paper proposes and axiomatizes a new updating rule: Relative Maximum Likelihood (RML) for ambiguous beliefs represented by a set of priors (C). This rule takes the form of applying Bayes' rule to a subset of C. This subset is a linear contraction of C towards its subset ascribing a maximal probability to the observed event. The degree of contraction captures the extent of willingness to discard priors based on likelihood when updating. Two well-known updating rules of multiple priors, Full Bayesian (FB) and Maximum Likelihood (ML), are included as special cases of RML. An axiomatic characterization of conditional preferences generated by RML updating is provided when the preferences admit Maxmin Expected Utility representations. The axiomatization relies on weakening the axioms characterizing FB and ML. The axiom characterizing ML is identified for the first time in this paper, addressing a long-standing open question in the literature.