Posterior Predictive Propensity Scores and $p$-Values

\citet{Rosenbaum83ps} introduced the notion of propensity score and discussed its central role in causal inference with observational studies. Their paper, however, causes 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 for 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.