Worst-case Optimal Probability Updating

This paper discusses an alternative to conditioning that may be used when the prob-ability distribution is not fully specified. It does not require any assumptions (suchas CAR: coarsening at random) on the unknown distribution. The well-known MontyHall problem is the simplest scenario where neither naive conditioning nor the CARassumption suffice to determine an updated probability distribution. This paper thusaddresses a generalization of that problem to arbitrary distributions on finite outcomespaces, arbitrary sets of ‘messages’, and (almost) arbitrary loss functions, and providesexistence and characterization theorems for worst-case optimal probability updatingstrategies. We find that for logarithmic loss, optimality is characterized by an elegantcondition, which we call RCAR (reverse coarsening at random). Under certain condi-tions, the same condition also characterizes optimality for a much larger class of lossfunctions, and we obtain an objective and general answer to how one should updateprobabilities in the light of new information.