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 measures, as a more realistic model of
uncertainty. Conditioning still makes sense in this context--we can simply
condition each measure in the set individually, then combine the results--and,
indeed, it seems to be the preferred 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 considerably longer (and not as
compelling) list of axioms that together force conditioning in this setting,
and describe other update methods that are allowed once any of these axioms is
dropped.
