Iterative importance sampling with Markov chain Monte Carlo sampling in robust Bayesian analysis

TL;DR

This paper introduces a new iterative importance sampling method tailored for robust Bayesian analysis that incorporates MCMC sampling, enabling efficient estimation of bounds on parameters amid prior uncertainty.

Contribution

It develops a novel approach to integrate MCMC sampling into iterative importance sampling for robust Bayesian inference, including a new effective sample size formula accounting for sample correlation.

Findings

Effective sample size expression accounts for MCMC correlation

Method successfully estimates bounds in meta-analysis example

Compared favorably to grid search under prior-data conflict

Abstract

Bayesian inference under a set of priors, called robust Bayesian analysis, allows for estimation of parameters within a model and quantification of epistemic uncertainty in quantities of interest by bounded (or imprecise) probability. Iterative importance sampling can be used to estimate bounds on the quantity of interest by optimizing over the set of priors. A method for iterative importance sampling when the robust Bayesian inference rely on Markov chain Monte Carlo (MCMC) sampling is proposed. To accommodate the MCMC sampling in iterative importance sampling, a new expression for the effective sample size of the importance sampling is derived, which accounts for the correlation in the MCMC samples. To illustrate the proposed method for robust Bayesian analysis, iterative importance sampling with MCMC sampling is applied to estimate the lower bound of the overall effect in a previously published meta-analysis with a random effects model. The performance of the method compared to a grid search method and under different degrees of prior-data conflict is also explored.