  We introduce a new updating rule, the conditional maximum likelihood rule
(CML) for updating ambiguous information. The CML formula replaces the
likelihood term in Bayes' rule with the maximal likelihood of the given signal
conditional on the state. We show that CML satisfies a new axiom, increased
sensitivity after updating, while other updating rules do not. With CML, a
decision maker's posterior is unaffected by the order in which independent
signals arrive. CML also accommodates recent experimental findings on updating
signals of unknown accuracy and has simple predictions on learning with such
signals. We show that an information designer can almost achieve her maximal
payoff with a suitable ambiguous information structure whenever the agent
updates according to CML.
