Robust Semiparametric Inference for Bayesian Additive Regression Trees

Abstract

We develop a corrected posterior distribution for semiparametric inference on the population mean under missing-at-random (MAR). The procedure combines Bayesian Additive Regression Trees (BART) with Bayesian-bootstrap reweighting. We derive a new Bernstein-von Mises (BvM) theorem and show that even the one-step posterior contains a bias term in the non-Donsker regime. To remove this term, we introduce RoBART, a posterior correction based on pilot estimators of the outcome regression and propensity score. We establish a BvM theorem for the corrected posterior and develop a cross-fitted version based on fold-specific BART posteriors. The average of fold-specific posterior means of RoBART coincides exactly with the corresponding cross-fitted augmented inverse-probability-weighted estimator, equivalently the double machine learning estimator. RoBART therefore provides a corrected posterior distribution for uncertainty quantification around the same point estimator. In simulations and an empirical illustration, RoBART demonstrates competitive finite-sample performance relative to existing methods.