To protect or not to protect: Bayes decisions with forecasts

A complete model, structural characterization, and analytic representation of the Bayes solution is presented for a protection problem: it is a sequential, stationary, finite-horizon, Markov decision process involving at each stage a dichotomous action, a binary state, and a one-period-ahead forecast of an independent (over time) Bernoulli input process. Two types of forecasts, categorical and probabilistic, are considered, along with their limiting cases, the naive and perfect forecasts. The concept of sufficiency is introduced and operationalized so that consistent comparisons of forecasts can be made with respect to both statistical and economic criteria. Mappings between the sufficiency ordering and the optimal decision strategy are proven or illustrated numerically.