Bayesian Matching for Causal Inference

In this paper we provide Bayesian matching methods for finding the causal effect of a binary intake variable x ∈ {0, 1} on an outcome of interest y. One technique we introduce is a Bayesian variant of the classic Rosenbaum and Rubin (1983, 1984) propensity score matching method. We show how it is possible to find the posterior distribution of the Bayesian matched sample average treatment effect (SATE) under weak semi-parametric assumptions regarding the model of the propensity score. We also consider another version of matching, which we call Bayesian self-matching, where the outcome distributions are modeled directly in a semi-parametric way and, for each subject, the expectation of the missing counterfactual (or the counterfactual itself) is obtained from the relevant outcome density. We also show how our Bayesian matching procedures generalize to randomized experiments with a compliance problem. Under the principal stratification environment of Frangakis and Rubin (2002), we show how to find the posterior distribution of the sample complier average treatment effect (SCATE) by matching on compliance probabilities and by Bayesian self-matching. We compare the Bayesian matching methods against each other and against frequentist matching estimators and causal estimates with both simulated and real data.