  Establishing the frequentist properties of Bayesian approaches widens their
appeal and offers new understanding. In hypothesis testing, Bayesian model
averaging addresses the problem that conclusions are sensitive to variable
selection. But Bayesian false discovery rate (FDR) guarantees are contingent on
prior assumptions that may be disputed. Here we show that Bayesian
model-averaged hypothesis testing is a closed testing procedure that controls
the frequentist familywise error rate (FWER) in the strong sense. The rate
converges pointwise as the sample size grows and, under some conditions,
uniformly. The `Doublethink' method computes simultaneous posterior odds and
asymptotic p-values for model-averaged hypothesis testing. We explore its
benefits, including post-hoc variable selection, and limitations, including
finite-sample inflation, through a Mendelian randomization study and
simulations comparing approaches like LASSO, stepwise regression, the
Benjamini-Hochberg procedure and e-values.
