Posterior Uncertainty for Targeted Parameters in Bayesian Bootstrap Procedures

TL;DR

This paper introduces a Bayesian method for targeted parameter inference in causal studies, ensuring valid uncertainty quantification even with nuisance parameters, and demonstrates its favorable theoretical properties.

Contribution

It develops a generalized Bayesian bootstrap approach for causal inference that maintains valid posterior inference with correct coverage, extending previous methods.

Findings

The proposed method has desirable frequentist properties.

Credible intervals achieve asymptotic correct coverage.

The approach is applicable to mis-specified and singly-robust models.

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.