Updating Probabilities
  As examples such as the Monty Hall puzzle show, applying conditioning to
update a probability distribution on a ``naive space', which does not take into
account the protocol used, can often lead to counterintuitive results. Here we
examine why. A criterion known as CAR (coarsening at random) in the statistical
literature characterizes when ``naive' conditioning in a naive space works. We
show that the CAR condition holds rather infrequently. We then consider more
generalized notions of update such as Jeffrey conditioning and minimizing
relative entropy (MRE). We give a generalization of the CAR condition that
characterizes when Jeffrey conditioning leads to appropriate answers, but show
that there are no such conditions for MRE. This generalizes and interconnects
previous results obtained in the literature on CAR and MRE.
