Ensemble Doubly Robust Bayesian Inference Via Regression Synthesis

The doubly robust estimator, which models both the propensity score and outcomes, is a popular approach to estimate the average treatment effect in the potential outcome setting. The primary appeal of this estimator is its theoretical property, wherein the estimator achieves consistency as long as either the propensity score or outcomes is correctly specified. In most applications, however, both are misspecified, leading to considerable bias that cannot be checked. In this paper, we propose a Bayesian ensemble approach that synthesizes multiple models for both the propensity score and outcomes, which we call doubly robust Bayesian regression synthesis. Our approach applies Bayesian updating to the ensemble model weights that adapt at the unit level, incorporating data heterogeneity, to significantly mitigate misspecification bias. Theoretically, we show that our proposed approach is consistent regarding the estimation of both the propensity score and outcomes, ensuring that the doubly robust estimator is consistent, even if no single model is correctly specified. An efficient algorithm for posterior computation facilitates the characterization of uncertainty regarding the treatment effect. Our proposed approach is compared against standard and state-of-the-art methods through two comprehensive simulation studies, where we find that our approach is superior in all cases. An empirical study on the impact of maternal smoking on birth weight highlights the practical applicability of our proposed method.