  We develop a semiparametric Bayesian approach for estimating the mean
response in a missing data model with binary outcomes and a nonparametrically
modelled propensity score. Equivalently we estimate the causal effect of a
treatment, correcting nonparametrically for confounding. We show that standard
Gaussian process priors satisfy a semiparametric Bernstein-von Mises theorem
under smoothness conditions. We further propose a novel propensity
score-dependent prior that provides efficient inference under strictly weaker
conditions. We also show that it is theoretically preferable to model the
covariate distribution with a Dirichlet process or Bayesian bootstrap, rather
than modelling the covariate density using a Gaussian process prior.
