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 t...