When ambiguous beliefs are represented by multiple priors, a decision maker (DM)'s updating rule may also include a step of refining the initial belief  . Maximum Likelihood (ML) updating indicates one example of such a refinement, whereas Full Bayesian (FB) updating does not allow for any refinement. In fact, ML and FB are the two extremes of refining beliefs with likelihoodin updating . To capture behaviors between these two extremes, the present paper proposes and axiomatizes a new updating rule , Relative Maximum Likelihood (RML) , to represent conditional preferences that are not as extreme as either case. RML updates an intermediate set of priors , which can be expressed as a linear contraction of the initial set with respect to the set of maximum likelihood priors . The linear contraction parameter captures the DM's relative inclination towards ML with respect to FB, it is also the threshold of a relative likelihood ratio test. Moreover, RML includes both FB and ML as the two extreme special cases . Additionally, the present paper provides an axiomatization of ML for preferences admit Maxmin Expected Utility (MEU) representation as a preliminary result for the exposition of RML .
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 FB are 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 .
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