Departing from Bayesian inference toward minimaxity to the extent that the posterior distribution is unreliable

Abstract

A Bayesian model may be relied on to the extent of its adequacy by minimizing the posterior expected loss raised to the power of a discounting exponent. The resulting action is minimax under broad conditions when the sample size is held fixed and the discounting exponent is infinite. On the other hand, for any finite discounting exponent, the action is Bayes when the sample size is sufficiently large. Thus, the action is Bayes when there is enough reliable information in the posterior distribution, is minimax when the posterior distribution is completely unreliable, and is a continuous blend of the two extremes otherwise.