Bayesian doubly robust causal inference via loss functions

Doubly robust causal inference has a well-established basis in frequentist semi-parametric theory, with estimation of causal parameters typically conducted via outcome regression and propensity score adjustment. A Bayesian counterpart, however, is not obvious as doubly robust estimation involves a semi-parametric formulation in the absence of a fully specified likelihood function. In this paper, we propose a Bayesian approach for doubly robust causal inference via two general Bayesian updating approaches based on loss functions. First, we specify a loss function for a doubly robust propensity score augmented outcome regression model and apply the traditional Bayesian updating mechanism which uses a prior belief distribution to calculate the posterior. Secondly, we draw inference for the posterior from a Bayesian predictive distribution via a Dirichlet process model, extending the Bayesian bootstrap. We show that these updating procedures yield valid posterior distributions of parameters which exhibit double robustness. Simulation studies show that the proposed methods can recover the true causal effect efficiently and achieve frequentist coverage even when the sample size is small or if the propensity score distribution is highly skewed. Finally, we apply our methods to evaluate the causal impact of speed cameras on traffic collisions in England.