Bayesian Propensity Score Estimators: Incorporating Uncertainties in Propensity Scores into Causal Inference

Despite their popularity, conventional propensity score estimators (PSEs) do not take into account uncertainties in propensity scores. This paper develops Bayesian propensity score estimators (BPSEs) to model the joint likelihood of both propensity score and outcome in one step, which naturally incorporates such uncertainties into causal inference. Simulations show that PSEs using estimated propensity scores tend to overestimate variations in the estimates of treatment effects—that is, too often they provide larger than necessary standard errors and lead to overly conservative inference—whereas BPSEs provide correct standard errors for the estimates of treatment effects and valid inference. Compared with other variance adjustment methods, BPSEs are guaranteed to provide positive standard errors, more reliable in small samples, can be readily employed to draw inference on individual treatment effects, etc. To illustrate the proposed methods, BPSEs are applied to evaluating a job training program. Accompanying software is available on the author's website.