  This paper discusses an alternative to conditioning that may be used when the
probability distribution is not fully specified. It does not require any
assumptions (such as CAR: coarsening at random) on the unknown distribution.
The well-known Monty Hall problem is the simplest scenario where neither naive
conditioning nor the CAR assumption suffice to determine an updated probability
distribution. This paper thus addresses a generalization of that problem to
arbitrary distributions on finite outcome spaces, arbitrary sets of `messages',
and (almost) arbitrary loss functions, and provides existence and
characterization theorems for robust probability updating strategies. We find
that for logarithmic loss, optimality is characterized by an elegant condition,
which we call RCAR (reverse coarsening at random). Under certain conditions,
the same condition also characterizes optimality for a much larger class of
loss functions, and we obtain an objective and general answer to how one should
update probabilities in the light of new information.
