Insensitivity to Non-Optimal Design in Bayesian Decision Theory

Abstract

Abstract Two simple inequalities involving two parameters, expected terminal losses (expected losses due to wrong decisions), expected sampling losses, and optimal (Bayes) and non-optimal sample sizes are shown to hold for several fixed sample size decision problems. The two parameters depend on the type of loss structure assumed. One inequality relates to the division of total expected losses for a sample of optimal size between expected terminal losses and expected sampling losses. The other inequality gives upper bounds on the ratio of total expected losses at non-optimal sample sizes to those at the optimal sample size. The latter inequality shows that total expected losses are often quite insensitive to the use of non-optimal sample sizes; in conjunction with optimal sample size formulas, it can be used to show that total expected losses are also insensitive to the use of a “wrong” prior distribution or the wrong cost parameters. The inequalities are shown to hold for several two-action problems on the mean of a Normal process, ten quadratic loss estimation problems involving Normal, Bernoulli, and Poisson processes, and one linear loss estimation problem on the mean of a Normal process; in each of these problems, a conjugate prior distribution is assumed. The inequalities are established by verifying two ad hoc conditions on expected terminal losses as a function of the sample size, which are proved to be sufficient for the inequalities. Optimal sample sizes for most of the problems considered have been given by Raiffa and Schlaifer [10].