  High-dimensional data can be useful for causal inference by providing many
confounders that may bolster the plausibility of the ignorability assumption.
Propensity score methods are powerful tools for causal inference, are popular
in health care research, and are particularly useful for high-dimensional data.
Recent interest has surrounded a Bayesian formulation of these methods in order
to flexibly estimate propensity scores and summarize posterior quantities while
incorporating variance from the (potentially high-dimensional) treatment model.
We discuss methods for Bayesian propensity score analysis of binary treatments,
focusing on modern methods for high-dimensional Bayesian regression and the
propagation of uncertainty from the treatment regression. We introduce a novel
and simple estimator for the average treatment effect that capitalizes on
conjugancy of the beta and binomial distributions. Through simulations, we show
the utility of horseshoe priors and Bayesian additive regression trees paired
with our new estimator, while demonstrating the importance of including
variance from the treatment and outcome models. Cardiac stent data with almost
500 confounders and 9000 patients illustrate approaches and compare among
existing frequentist alternatives.
