The Central Role of Bayes’ Theorem for Joint Estimation of Causal Effects and Propensity Scores

Abstract

Although propensity scores have been central to the estimation of causal effects for over 30 years, only recently has the statistical literature begun to consider in detail methods for Bayesian estimation of propensity scores and causal effects. Underlying this recent body of literature on Bayesian propensity score estimation is an implicit discordance between the goal of the propensity score and the use of Bayes theorem. The propensity score condenses multivariate covariate information into a scalar to allow estimation of causal effects without specifying a model for how each covariate relates to the outcome. Avoiding specification of a detailed model for the outcome response surface is valuable for robust estimation of causal effects, but this strategy is at odds with the use of Bayes theorem, which presupposes a full probability model for the observed data that adheres to the likelihood principle. The goal of this paper is to explicate this fundamental feature of Bayesian estimation of causal effects with propensity scores in order to provide context for the existing literature and for future work on this important topic.

Causal Inference with Longitudinal Outcomes and Non-Ignorable Dropout: Estimating the Effect of Living Alone on Cognitive Decline

“…Then, conditionally on each matched set, we fit the mixed model. Note that we are not using a joint likelihood as the two components above do not factor (Zigler, 2013); instead we are doing the inference in two stages to account appropriately for uncertainty in the matching (and to avoid feedback) as was recommended in Zigler (2013).…”

Bayesian estimation of the average treatment effect on the treated using inverse weighting

“…Following the works of Kaplan and Chen and Zigler, we now consider a more complex structure for the treatment model. More specifically, we assume Z i ∼ Bern ( e i ), where , where X 1 ∼ N (1,1), X 2 ∼ Poisson (2), X 3 ∼ Bernoulli (0.5), X 4 ∼ N (0,1), X 5 ∼ N (1,1), X 6 ∼ N (0,1), and | X 4 | represents the absolute value of X 4 .…”

“…In contrast, Bayesian procedures typically perform estimation and inference by considering a joint likelihood function for all parameters of interest (in this case, the exposure and outcome), so that the uncertainty in the modeling of the propensity score is propagated into the estimation of the treatment effect. While there has been relatively little literature on Bayesian propensity score approaches to causal inference, that literature has called into question the traditional joint modeling approach (see the works of Rubin and Zigler for a review).…”

Bayesian propensity scores for high‐dimensional causal inference: A comparison of drug‐eluting to bare‐metal coronary stents

“…We do not offer new theory on Bayesian propensity scores, referring the interested reader to the extensive discussion in Saarela et al. () and its accompanying critiques, as well as Zigler (). Our interest is on the potential utility of a two‐step Bayesian propensity score framework in high‐dimensional causal inference problems.…”

“…By focusing on a Bayesian framework, we allow for intuitive and flexible approaches to model building with many variables while propagating uncertainty in the treatment model. We do not offer new theory on Bayesian propensity scores, referring the interested reader to the extensive discussion in Saarela et al (2015) and its accompanying critiques, as well as Zigler (2016). Our interest is on the potential utility of a two-step Bayesian propensity score framework in high-dimensional causal inference problems.…”