Introduction to and Recent Developments in Bayesian Causal Inference: Propensity Scores, Treatment Effect Heterogeneity, and Sensitivity Analysis

In recent years, Bayesian approaches to causal inference have advanced substantially and attracted increasing attention. This article provides an overview of Bayesian methods for causal inference, with a focus on estimating causal effects for point treatments. As an introduction to Bayesian causal inference, we first revisited causal identification based on the g-formula and clarified how Bayesian inference could be linked to observed data likelihood under standard identifying assumptions. We also organized the role of the propensity score in Bayesian causal inference. We then reviewed Bayesian modeling tools commonly used as flexible outcome models, including nonparametric Bayesian models such as Gaussian processes, Bayesian additive regression trees, and Dirichlet process mixture models, as well as generalized Bayesian inference, which enabled model-free Bayesian updating. Subsequently, we discussed three recent developments in Bayesian causal inference. First, we examined the propensity score and feedback problem, highlighting the challenges that arose when propensity scores were incorporated into Bayesian inference, and reviewed possible solutions. Second, we reviewed methods for estimating treatment effect heterogeneity, with particular emphasis on Bayesian causal forests. Third, we introduced two approaches to sensitivity analysis, which represented a key advantage of Bayesian inference.