  We introduce two kinds of risk measures with respect to some reference
probability measure, which both allow for a certain order structure and
domination property. Analyzing their relation to each other leads to the
question when a certain minimax inequality is actually an equality. We then
provide conditions under which the corresponding robust risk measures, being
defined as the supremum over all risk measures induced by a set of probability
measures, can be represented classically in terms of one single probability
measure. We focus in particular on the mixture probability measure obtained via
mixing over a set of probability measures using some prior, which represents
for instance the regulator's beliefs. The classical representation in terms of
the mixture probability measure can then be interpreted as a Bayesian approach
to robust risk measures.
