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.

翻译：在常态半参数理论中,稳健的因果关系推断有一个牢固的既定基础,对因果参数的估计通常是通过结果回归和偏差分分分的调整来进行的。不过,巴耶斯对口方的估算并不明显,因为双重稳健的估算涉及在没有完全具体的可能性函数情况下的半参数配方。在本文中,我们建议采用巴伊西亚方法,通过基于损失功能的两种通用巴耶斯更新方法进行双稳健的因果推断。首先,我们为双稳健的常态评分确定一个损失函数,增加结果回归模型,并采用传统的巴伊西亚更新机制,即使用先前的信仰分布来计算后方。第二,我们从巴伊斯的预测分布中推断出通过迪里特进程模型进行半参数配方的推论,以延长贝耶斯河靴区。我们表明,这些更新程序产生有效的后方参数分布结果显示双强力。模拟研究表明,拟议的方法可以有效恢复真实的因果效应,并实现常态覆盖,即使样本规模小,或者误差分点分的英格兰碰撞速度。最后,我们用方法评估了英格兰的交通碰撞影响。