Asymptotics of cut distributions and robust modular inference using Posterior Bootstrap

TL;DR

This paper analyzes the asymptotic properties of cut distributions in Bayesian inference, introduces a Posterior Bootstrap algorithm for robust inference, and demonstrates its effectiveness through numerical experiments including causal inference.

Contribution

It provides the first Bernstein-von Mises theorem and Laplace approximation for cut distributions, and proposes a Posterior Bootstrap method for credible regions with correct frequentist coverage.

Findings

Bernstein-von Mises theorem for cut distributions

Laplace approximation with quantitative bounds

Posterior Bootstrap achieves nominal frequentist coverage

Abstract

Bayesian inference provides a framework to combine various model components with shared parameters, allowing joint uncertainty estimation and the use of all available data sources. Unfortunately, misspecification of any part of the model might propagate to all other parts and can lead to unsatisfactory results. Cut distributions have been proposed as a remedy, where the information is prevented from flowing along certain directions. We study cut distributions from an asymptotic perspective and obtain a Bernstein-von Mises theorem, as well as a Laplace approximation with quantitative bounds. We then propose an algorithm based on the Posterior Bootstrap that delivers credible regions with the nominal frequentist asymptotic coverage. The proposed methods are illustrated with numerical experiments in a variety of examples, including causal inference with propensity scores.