Double Robust Bayesian Inference on Average Treatment Effects

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.