Bayesian model-averaged hypothesis testing is an important technique in regression because it addresses the problem that the evidence one variable directly affects an outcome often depends on which other variables are included in the model. This problem is caused by confounding and mediation, and is pervasive in big data settings with thousands of variables. However, model-averaging is under-utilized in fields, like epidemiology, where classical statistical approaches dominate. Here we show that simultaneous Bayesian and frequentist model-averaged hypothesis testing is possible in large samples, for a family of priors. We show that Bayesian model-averaged regression is a closed testing procedure, and use the theory of regular variation to derive interchangeable posterior odds and $p$-values that jointly control the Bayesian false discovery rate (FDR), the frequentist type I error rate, and the frequentist familywise error rate (FWER). These results arise from an asymptotic chi-squared distribution for the model-averaged deviance, under the null hypothesis. We call the approach 'Doublethink'. In a related manuscript (Arning, Fryer and Wilson, 2024), we apply it to discovering direct risk factors for COVID-19 hospitalization in UK Biobank, and we discuss its broader implications for bridging the differences between Bayesian and frequentist hypothesis testing.

翻译：贝叶斯模型平均假设检验是回归分析中的重要技术，因为它解决了这样一个问题：某个变量直接影响结果的证据往往取决于模型中包含的其他变量。这一现象由混杂和中介效应引起，在包含数千个变量的大数据环境中普遍存在。然而，在流行病学等以经典统计方法为主导的领域中，模型平均技术尚未得到充分利用。本文证明，对于一类先验分布，在大样本条件下可以同时进行贝叶斯和频率学派的模型平均假设检验。研究表明，贝叶斯模型平均回归是一种闭合检验程序，并利用正则变化理论推导出可互换的后验赔率与$p$值，这些指标共同控制了贝叶斯错误发现率（FDR）、频率学派第一类错误率以及频率学派家庭误差率（FWER）。这些结果源于零假设下模型平均偏差的渐近卡方分布。我们将该方法称为"双重思维"。在相关研究手稿（Arning, Fryer and Wilson, 2024）中，我们将其应用于发现UK Biobank中COVID-19住院的直接风险因素，并讨论了该方法在弥合贝叶斯与频率学派假设检验差异方面的广泛意义。