We study a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our Bayesian approach involves a correction term for prior distributions adjusted by the propensity score. We prove asymptotic equivalence of our Bayesian estimator and efficient frequentist estimators by establishing a new semiparametric Bernstein-von Mises theorem under double robustness; i.e., the lack of smoothness of conditional mean functions can be compensated by high regularity of the propensity score and vice versa. Consequently, the resulting Bayesian point estimator internalizes the bias correction as the frequentist-type doubly robust estimator, and the Bayesian credible sets form confidence intervals with asymptotically exact coverage probability. In simulations, we find that this corrected Bayesian procedure leads to significant bias reduction of point estimation and accurate coverage of confidence intervals, especially when the dimensionality of covariates is large relative to the sample size and the underlying functions become complex. We illustrate our method in an application to the National Supported Work Demonstration.

翻译：我们研究了一种关于平均治疗效果(ATE)的双倍有力的贝叶斯推断程序。我们的贝叶斯方法涉及对先前经倾向性评分调整的分布进行校正的术语。我们通过在双重强势下建立新的半参数Bernstein-von Mises 理论,证明我们的贝叶斯天顶和高效的常客测算器的无症状等同性。我们发现,这种校正的贝叶斯程序导致点估计和准确信任期的偏差显著减少,特别是当共变的尺寸与样本大小和基本功能的复杂程度相比较大时。我们用常态型双型稳健的估测算器将偏差修正内部化为常态,而巴伊斯人可信的定型则形成信任间隔,其覆盖概率是无症状的精确概率。在模拟中,我们发现,这种校正的贝伊斯程序导致点估计和准确度间隔的偏差显著降低,特别是当共变的尺寸与样本大小和基本功能变得复杂时。我们用在应用国家支持性演示时说明我们的方法。