Double Robust Bayesian Inference on Average Treatment Effects

We study a double robust Bayesian inference procedure on the average treatment effect (ATE) under unconfoundedness. Our Bayesian approach involves a correction term for prior distributions adjusted by the propensity score. We prove asymptotic equivalence of our Bayesian estimator and efficient frequentist 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 point estimator internalizes the bias correction as the frequentist-type doubly robust estimator, and the Bayesian credible sets form confidence intervals with asymptotically exact coverage probability. In simulations, we find that this corrected Bayesian procedure leads to significant bias reduction of point estimation and accurate coverage of confidence intervals, especially when the dimensionality of covariates is large relative to the sample size and the underlying functions become complex. We illustrate our method in an application to the National Supported Work Demonstration.

Adaptive Estimation of Quadratic Functionals in Nonparametric Instrumental Variable Models

This paper considers adaptive estimation of quadratic functionals in the nonparametric instrumental variables (NPIV) models. Minimax estimation of a quadratic functional of a NPIV is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator using one random sample. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional proposed by \cite{BC2020} attains a convergence rate that coincides with the lower bound previously derived by \cite{ChenChristensen2017}. The minimax rate is achieved by the optimal choice of a key tuning parameter (sieve dimension) that depends on unknown NPIV model features. We next propose a data driven choice of the tuning parameter based on Lepski's method. The adaptive estimator attains the minimax optimal rate in the severely ill-posed case and in the regular, mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ in the irregular, mildly ill-posed case.