Semiparametric posterior corrections

We present a new approach to semiparametric inference using corrected posterior distributions. The method allows us to leverage the adaptivity, regularization, and predictive power of nonparametric Bayesian procedures to estimate low-dimensional functionals of interest without being restricted by the holistic Bayesian formalism. Starting from a conventional posterior on the whole data-generating distribution, we correct the marginal posterior for each functional of interest with the help of the Bayesian bootstrap. We provide conditions for the resulting one-step posterior to possess calibrated frequentist properties and specialize the results for several canonical examples: the integrated squared density, the mean of a missing-at-random outcome, and the average causal treatment effect on the treated. The procedure is computationally attractive, requiring only a simple, efficient postprocessing step that can be attached onto any arbitrary posterior sampling algorithm. Using the ACIC 2016 causal data analysis competition, we illustrate that our approach can outperform the existing state-of-the-art through the propagation of Bayesian uncertainty.