Bayesian sensitivity analysis for a missing data model

In causal inference, it is important to study the sensitivity of the conclusions to key assumptions. We perform sensitivity analysis of the assumption that missing outcomes are missing completely at random. We follow a Bayesian approach, which is nonparametric for the outcome distribution and can be combined with an informative prior on the sensitivity parameter. We give insight in the posterior and provide theoretical guarantees in the form of Bernstein-von Mises theorems for estimating the mean outcome. We study different parametrisations of the model involving Dirichlet process priors on the distribution of the outcome and on the distribution of the outcome conditional on the subject being treated. We show that these parametrisations incorporate a prior on the sensitivity parameter in different ways and discuss the relative merits. A key result on which the above theorems rely will be a general Bernstein-von Mises theorem for the normalised extended gamma process. We also present a simulation study, showing the performance of the methods in finite sample scenarios.