In the case where beliefs are ambiguous and represented by multiple priors, a decision maker (DM)'s updating behavior may also include a revision and refinement of her initial belief, a process which is absent from the Bayesian updating. As known in the literature, Maximum Likelihood (ML) updating provides one method of such refinement. The present paper provides an axiomatization of ML for preferences admit Maxmin Expected Utility (MEU) representation. ML and Full Bayesian (FB ) are two updating rules with polar opposite methods of refining ambiguous beliefs . The present paper proposes and axiomatizes a new updating rule, Relative Maximum Likelihood (RML), for when a DM's conditional preference is not as extreme as either case. RML updates  a linear contraction of the set of priors with respect to the  maximum likelihood priors. The linear contraction parameter captures the relative attitudes of the DM towards ML with respect to FB . In particular, when the parameter takes either one of the two extreme values (0 or 1), RML reduces to FB or ML respectively. This paper also characterizes the Extended RML updating rule for ex-ante preferences that in addition satisfy comonotonic independence. This rule, which is related to RML, further preserves comonotonic independence for the updated preferences
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 likelihood in 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.
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