  We consider a setting where an agent's uncertainty is represented by a set of
probability measures, rather than a single measure. Measure-by-measure updating
of such a set of measures upon acquiring new information is well-known to
suffer from problems; agents are not always able to learn appropriately. To
deal with these problems, we propose using weighted sets of probabilities: a
representation where each measure is associated with a weight, which denotes
its significance. We describe a natural approach to updating in such a
situation and a natural approach to determining the weights. We then show how
this representation can be used in decision-making, by modifying a standard
approach to decision making -- minimizing expected regret -- to obtain minimax
weighted expected regret (MWER). We provide an axiomatization that
characterizes preferences induced by MWER both in the static and dynamic case.
