Double Robust Bayesian Inference on Average Treatment Effects

TL;DR

This paper introduces a double robust Bayesian method for estimating the average treatment effect that combines prior adjustments and posterior correction, achieving asymptotic efficiency and accurate confidence intervals.

Contribution

It develops a novel Bayesian inference procedure for ATE that is asymptotically equivalent to efficient frequentist estimators under double robustness.

Findings

Provides precise ATE point estimates and credible intervals.

Achieves shorter confidence intervals compared to existing methods.

Demonstrates effectiveness in simulation and real data applications.

Abstract

We propose a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. For our new Bayesian approach, we first adjust the prior distributions of the conditional mean functions, and then 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 method provides precise point estimates of the ATE through the posterior mean and credible intervals that closely align with the nominal coverage probability. Furthermore, our approach achieves a shorter interval length in comparison to existing methods. We illustrate our method in an application to the National Supported Work Demonstration following LaLonde [1986] and Dehejia and Wahba [1999].