On Bayes factors for the linear model

We show that the Bayes factor for testing whether a subset of coefficients are zero in the normal linear regression model gives the uniformly most powerful test amongst the class of invariant tests discussed in Lehmann & Romano (2005) if the prior distributions for the regression coefficients are in a specific class of distributions. The priors in this class can have any elliptical distribution, with a specific scale matrix, for the subset of coefficients that are being tested. We also show under mild conditions that the Bayes factor gives the uniformly most powerful invariant test only if the prior for the coefficients being tested is an elliptical distribution with this scale matrix. The implications are discussed.