We study semiparametric Bayesian estimation of the mean response in a binary regression model with missing observations. We allow some dependence between the missingness and response mechanisms, which we assume are conditionally independent given some measured covariates (i.e. unconfoundedness). This model has applications in biostatistics and causal inference. We show that the marginal posterior distribution for the mean response arising from product priors on the different model parameters satisfies a semiparametric Bernstein-von Mises theorem under some conditions. We also propose a more involved prior geared towards estimating this specific functional.
This is joint work with Aad van der Vaart.