Posterior Uncertainty for Targeted Parameters in Bayesian Bootstrap Procedures
Magid Sabbagh, David A. Stephens
TL;DR
The paper addresses valid Bayesian inference for finite-dimensional targeted parameters in causal models when a nuisance parameter is present. It develops and analyzes the Linked Bayesian Bootstrap, a Dirichlet-weighted, estimating-equation–driven approach that propagates nuisance-parameter uncertainty into the posterior for the targeted parameter, while also offering a fully Bayesian one-step interpretation. It proves asymptotic Bayesian–frequentist duality and consistency results, and shows credible intervals achieve nominal coverage; it also contrasts with conventional plug-in methods that inflate posterior variance. Simulations on G-estimation and weighting estimators illustrate that posterior means align with frequentist targets and that coverage remains near nominal, validating the method’s practical reliability. Overall, the work provides a principled, nonparametric Bayesian framework for causal inference under misspecification and estimating-equation settings, with broad implications for Bayesian nonparametric inference in causal analysis.
Abstract
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.