We propose a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our robust Bayesian approach involves two important modifications: first, we adjust the prior distributions of the conditional mean function; second, we correct the posterior distribution of the resulting ATE. Both adjustments make use of pilot estimators motivated by the semiparametric influence function for ATE estimation. We prove asymptotic equivalence of our Bayesian procedure and efficient frequentist ATE 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 credible sets form confidence intervals with asymptotically exact coverage probability. In simulations, our double robust Bayesian procedure leads to significant bias reduction of point estimation over conventional Bayesian methods and more accurate coverage of confidence intervals compared to existing frequentist methods. We illustrate our method in an application to the National Supported Work Demonstration.

翻译：我们提出了一种在无混杂假设下对平均处理效应进行双稳健贝叶斯推断的方法。该方法包含两项重要改进：其一，调整条件均值函数的先验分布；其二，修正所得平均处理效应的后验分布。两项调整均利用了基于半参数影响函数的平均处理效应估计引导估计量。通过建立双稳健条件下新的半参数伯恩斯坦-冯·米塞斯定理，我们证明了贝叶斯方法与高效频率学派平均处理效应估计量的渐近等价性——即条件均值函数光滑性的不足可由倾向得分的高正则性补偿，反之亦然。由此构建的贝叶斯可信集可作为置信区间使用，并具有渐近精确的覆盖概率。在模拟实验中，与传统贝叶斯方法相比，我们的双稳健贝叶斯方法显著降低了点估计偏差；与现有频率学派方法相比，其置信区间覆盖更为精确。最后，我们将该方法应用于美国国家支持工作示范项目的数据分析。