We propose a general method to carry out a valid Bayesian analysis of a finite-dimensional `targeted' parameter in the presence of a finite-dimensional nuisance parameter. We apply our methods to causal inference based on estimating equations. While much of the literature in Bayesian causal inference has relied on the conventional 'likelihood times prior' framework, a recently proposed method, the 'Linked Bayesian Bootstrap', deviated from this classical setting to obtain valid Bayesian inference using the Dirichlet process and the Bayesian bootstrap. These methods rely on an adjustment based on the propensity score and explain how to handle the uncertainty concerning it when studying the posterior distribution of a treatment effect. We examine theoretically the asymptotic properties of the posterior distribution obtained and show that our proposed method, a generalized version of the 'Linked Bayesian Bootstrap', enjoys desirable frequentist properties. In addition, we show that the credible intervals have asymptotically the correct coverage properties. We discuss the applications of our method to mis-specified and singly-robust models in causal inference.

翻译：本文提出了一种通用方法，用于在存在有限维干扰参数的情况下对有限维"目标"参数进行有效的贝叶斯分析。我们将该方法应用于基于估计方程的因果推断研究。虽然贝叶斯因果推断领域的大量文献依赖于传统的"似然乘以先验"框架，但最近提出的"关联贝叶斯自助法"突破了这一经典设定，通过狄利克雷过程和贝叶斯自助法获得了有效的贝叶斯推断。这些方法基于倾向得分进行调整，并阐释了在研究处理效应的后验分布时如何处理相关的不确定性。我们从理论上检验了所得后验分布的渐近性质，证明我们提出的方法——即"关联贝叶斯自助法"的广义版本——具备理想的频率学派性质。此外，我们证明了可信区间具有渐近正确的覆盖特性。最后，我们讨论了该方法在因果推断中误设模型和单稳健模型中的应用。