Semiparametric Bayesian causal inference

We develop a semiparametric Bayesian approach for estimatingthe mean response in a missing data model with binary outcomesand a nonparametrically modelled propensity score. Equivalently, weestimate the causal effect of a treatment, correcting nonparamet-rically for confounding. We show that standard Gaussian processpriors satisfy a semiparametric Bernstein–von Mises theorem undersmoothness conditions. We further propose a novel propensity score-dependent prior that provides efficient inference under strictly weakerconditions. We also show that it is theoretically preferable to modelthe covariate distribution with a Dirichlet process or Bayesian boot-strap, rather than modelling its density