We study a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our robust Bayesian approach involves two adjustment steps: first, we make a correction for prior distributions of the conditional mean function; second, we introduce a recentering term on the posterior distribution of the resulting ATE. We prove asymptotic equivalence of our Bayesian estimator and double robust 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 robust 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：本文研究了在无混淆假设下，针对平均处理效应（ATE）的双稳健贝叶斯推断过程。我们的稳健贝叶斯方法包含两个调整步骤：首先，对条件均值函数的先验分布进行修正；其次，在得到的ATE后验分布中引入一个重新中心化项。通过建立一个新的满足双稳健性的半参数伯恩斯坦-冯·米塞斯定理，我们证明了所提出的贝叶斯估计量与双稳健频率学派估计量在渐近意义下等价；即，条件均值函数的光滑性不足可由倾向得分的高度正则性补偿，反之亦然。因此，所得到的贝叶斯点估计量内化了类似频率学派双稳健估计量的偏差校正，而贝叶斯可信集构成的置信区间具有渐近精确的覆盖概率。在模拟中，我们发现该稳健贝叶斯过程能显著降低点估计的偏差，并提供准确的置信区间覆盖，尤其在协变量维度相对于样本量较大且底层函数变得复杂时。我们将该方法应用于美国国家支持工作示范项目中的实例分析。