Restricted most powerful Bayesian tests for linear models

Uniformly most powerful Bayesian tests (UMPBTs) are a new class of Bayesian tests in which null hypotheses are rejected if their Bayes factor exceeds a specified threshold. The alternative hypotheses in UMPBTs are defined to maximize the probability that the null hypothesis is rejected. Here, we generalize the notion of UMPBTs by restricting the class of alternative hypotheses over which this maximization is performed, resulting in restricted most powerful Bayesian tests (RMPBTs). We then derive RMPBTs for linear models by restricting alternative hypotheses to g priors. For linear models, the rejection regions of RMPBTs coincide with those of usual frequentist F-tests, provided that the evidence thresholds for the RMPBTs are appropriately matched to the size of the classical tests. This correspondence supplies default Bayes factors for many common tests of linear hypotheses. We illustrate the use of RMPBTs for ANOVA tests and t-tests and compare their performance in numerical studies.