When ambiguous beliefs are represented by multiple priors, updating such a belief may also include refining of the initial belief by discarding some of the priors. Maximum Likelihood (ML) and Full Bayesian (FB) are the two extremes of refining beliefs based on likelihood. To capture behaviors between these two extremes, this paper proposes a new updating rule: Relative Maximum Likelihood (RML) updating . This rule updates an intermediate set of priors that is a linear contraction of the initial set with respect to the set of priors attaining maximum likelihood. The linear contraction parameter captures the DM's relative inclination towards ML with respect to FB, hence ML and FBare the two extreme special cases of RML. More importantly, characterization of conditional preferences given by RML updating is provided when the preferences are represented by the Maxmin Expected Utility  . The axiomatization relies on weakening the axioms that characterize FB and ML. In particular, the axiom that characterizes ML is identified  in this paper, fixing an existing problem in the literature.
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 the set C. This subset is a linear contraction of C towards its subset assigning 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.
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