  We propose a doubly robust approach to characterizing treatment effect
heterogeneity in observational studies. We develop a frequentist inferential
procedure that utilizes posterior distributions for both the propensity score
and outcome regression models to provide valid inference on the conditional
average treatment effect even when high-dimensional or nonparametric models are
used. We show that our approach leads to conservative inference in finite
samples or under model misspecification, and provides a consistent variance
estimator when both models are correctly specified. In simulations, we
illustrate the utility of these results in difficult settings such as
high-dimensional covariate spaces or highly flexible models for the propensity
score and outcome regression. Lastly, we analyze environmental exposure data
from NHANES to identify how the effects of these exposures vary by
subject-level characteristics.
