Updating Probabilities

As examples such as the Monty Hall puzzle show, applying conditioning to update a proba­ bility distribution on a "naive space", which does not take into account the protocol used, can often lead to counterintuitive results. Here we exam­ ine why. A criterion known as CAR ("coarsening at random") in the statistical literature character­ izes when "naive" conditioning in a naive space works. We show that the CAR condition holds rather infrequently. We then consider more gen­ eralized notions of update such as Jeffrey condi­ tioning and minimizing relative entropy (MRE). We give a generalization of the CAR condi­ tion 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.