  Although propensity scores have been central to the estimation of causal
effects for over 30 years, only recently has the statistical literature begun
to consider in detail methods for Bayesian estimation of propensity scores and
causal effects. Underlying this recent body of literature on Bayesian
propensity score estimation is an implicit discordance between the goal of the
propensity score and the use of Bayes theorem. The propensity score condenses
multivariate covariate information into a scalar to allow estimation of causal
effects without specifying a model for how each covariate relates to the
outcome. Avoiding specification of a detailed model for the outcome response
surface is valuable for robust estimation of causal effects, but this strategy
is at odds with the use of Bayes theorem, which presupposes a full probability
model for the observed data. The goal of this paper is to explicate this
fundamental feature of Bayesian estimation of causal effects with propensity
scores in order to provide context for the existing literature and for future
work on this important topic.
