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Estimating causal effects with propensity scores relies upon the availabilityof treated and untreated units observed at each value of the estimatedpropensity score. In settings with strong confounding, limited so-called"overlap" in propensity score distributions can undermine the empirical basisfor estimating causal effects and yield erratic finite-sample performance ofexisting estimators. We propose a Bayesian procedure designed to estimatecausal effects in settings where there is limited overlap in propensity scoredistributions. Our method relies on the posterior predictive treatmentassignment (PPTA), a quantity that is derived from the propensity score butserves different role in estimation of causal effects. We use the PPTA toestimate causal effects by marginalizing over the uncertainty in whether eachobservation is a member of an unknown subset for which treatment assignment canbe assumed unconfounded. The resulting posterior distribution depends on theempirical basis for estimating a causal effect for each observation and hascommonalities with recently-proposed "overlap weights" of Li et al. (2016). Weshow that the PPTA approach can be construed as a stochastic version ofexisting ad-hoc approaches such as pruning based on the propensity score ortruncation of inverse probability of treatment weights, and highlight severalpractical advantages including uncertainty quantification and improvedfinite-sample performance. We illustrate the method in an evaluation of theeffectiveness of technologies for reducing harmful pollution emissions frompower plants in the United States.

Posterior Predictive Treatment Assignment for Estimating Causal Effects with Limited Overlap