Posterior Uncertainty for Targeted Parameters in Bayesian Bootstrap Procedures

2026-02-02

AI Summary

This study addresses the challenge of conducting valid Bayesian inference for target parameters—such as causal effects—in the presence of finite-dimensional nuisance parameters. The authors propose a general framework that integrates Bayesian bootstrap with Dirichlet process priors within an estimating equation approach, explicitly accounting for uncertainty in propensity score estimation. The method is robust to model misspecification and remains valid under only single robustness conditions. It extends the "linked Bayesian bootstrap" to nonstandard Bayesian settings, yielding posterior inferences with favorable frequentist properties. Theoretical analysis demonstrates that the resulting posterior distribution exhibits desirable asymptotic behavior, with credible intervals achieving nominal coverage probabilities in large samples.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Bayesian bootstrap
• causal inference
• targeted parameter
• nuisance parameter
• propensity score

Innovation

• Bayesian bootstrap
• targeted parameter
• causal inference
• propensity score
• frequentist properties