  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.
