Updating Sets of Probabilities

There are several well-known justifications for conditioning as the appropriate method for updating a single probability measure, given an observation. However, there is a significant body of work arguing for sets of probability measures, rather than single mea­ sures, as a more realistic model of uncer­ tainty. Conditioning still makes sense in this context-we can simply condition each mea­ sure in the set individually, then combine the results-and, indeed, it seems to be the pre� ferred updating procedure in the literature. But how justified is conditioning in this richer setting? Here we show, by considering an axiomatic account of conditioning given by van Fraassen, that the single-measure and sets-of-measures cases are very different. We show that van Fraassen's axiomatization for the former case is nowhere near sufficient for updating sets of measures. We give a con­ siderably longer (and not as compelling) list of axioms that together force conditioning in this setting, and describe other update meth­ ods that are allowed once any of these axioms is dropped.