  We study Bayesian approaches to causal inference via propensity score
regression. Much of the Bayesian literature on propensity score methods have
relied on approaches that cannot be viewed as fully Bayesian in the context of
conventional `likelihood times prior' posterior inference; in addition, most
methods rely on parametric and distributional assumptions, and presumed correct
specification. We emphasize that causal inference is typically carried out in
settings of mis-specification, and develop strategies for fully Bayesian
inference that reflect this. We focus on methods based on decision-theoretic
arguments, and show how inference based on loss-minimization can give valid and
fully Bayesian inference. We propose a computational approach to inference
based on the Bayesian bootstrap which has good Bayesian and frequentist
properties.
