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.

翻译：确立贝叶斯方法的频率主义性质能拓宽其吸引力并提供新的理解。在假设检验中，贝叶斯模型平均解决了结论对变量选择敏感的问题。但贝叶斯错误发现率（FDR）的保证依赖于可能存疑的先验假设。本文证明，贝叶斯模型平均假设检验是一种闭包检验程序，能在强控制意义上控制频率主义族错误率（FWER）。该速率随样本量增长逐点收敛，并在某些条件下一致收敛。“双重思维”方法为模型平均假设检验计算同步后验概率比与渐近p值。我们通过一项孟德尔随机化研究和模拟实验（比较LASSO、逐步回归、Benjamini-Hochberg程序及e值等方法），探讨了其优势（包括事后变量选择）与局限（包括有限样本膨胀问题）。