  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.
