Abstract: We study belief revision when information is given as a set of relevant probability distributions. This flexible setting encompasses (i) the standard notion of information as an event (a subset of the state space), (ii) qualitative information (``A is more likely than B"), (iii) interval information (``chance of A is between ten and twenty percent"), and more. In this setting, we behaviorally characterize a decision maker (DM) who selects a posterior belief from the provided information set that minimizes the subjective distance between her prior and the information. We call such a DM a Minimum Distance Subjective Expected Utility (MDSEU) maximizer. Next, we characterize the collection of MDSEU distance notions that coincide with Bayesian updating on standard events. We call this class of distances Generalized Bayesian Divergence, as they nest Kullback-Leibler Divergence. MDSEU provides a systematic way to extend Bayesian updating to general information and zero-probability events. Importantly, Bayesian updating is not unique. Thus, two Bayesian DM's with a common prior may disagree after common information, resulting in polarization and speculative trade. We discuss related models of non-Bayesian updating.

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