  In observational studies, causal inference relies on several key identifying
assumptions. One identifiability condition is the positivity assumption, which
requires the probability of treatment be bounded away from 0 and 1. That is,
for every covariate combination, it should be possible to observe both treated
and control subjects, i.e., the covariate distributions should overlap between
treatment arms. If the positivity assumption is violated, population-level
causal inference necessarily involves some extrapolation. Ideally, a greater
amount of uncertainty about the causal effect estimate should be reflected in
such situations. With that goal in mind, we construct a Gaussian process model
for estimating treatment effects in the presence of practical violations of
positivity. Advantages of our method include minimal distributional
assumptions, a cohesive model for estimating treatment effects, and more
uncertainty associated with areas in the covariate space where there is less
overlap. We assess the performance of our approach with respect to bias and
efficiency using simulation studies. The method is then applied to a study of
critically ill female patients to examine the effect of undergoing right heart
catheterization.
