  \citet{Rosenbaum83ps} introduced the notion of the propensity score and
discussed its central role in causal inference with observational studies.
Their paper, however, caused a fundamental incoherence with an early paper by
\citet{Rubin78}, which showed that the propensity score does not play any role
in the Bayesian analysis of unconfounded observational studies if the priors on
the propensity score and outcome models are independent. Despite the serious
efforts made in the literature, it is generally difficult to reconcile these
contradicting results. We offer a simple approach to incorporating the
propensity score in Bayesian causal inference based on the posterior predictive
$p$-value. To motivate a simple procedure, we focus on the model with the
strong null hypothesis of no causal effects for any units whatsoever.
Computationally, the proposed posterior predictive $p$-value equals the classic
$p$-value based on the Fisher randomization test averaged over the posterior
predictive distribution of the propensity score. Moreover, using the
studentized doubly robust estimator as the test statistic, the proposed
$p$-value inherits the doubly robust property and is also asymptotically valid
for testing the weak null hypothesis of zero average causal effect. Perhaps
surprisingly, this Bayesianly motivated $p$-value can have better frequentist's
finite-sample performance than the frequentist's $p$-value based on the
asymptotic approximation especially when the propensity scores can take extreme
values.
