Post-Processing of MCMC

TL;DR

This paper reviews advanced techniques for post-processing MCMC outputs, focusing on methods that address bias-variance trade-offs and improve the accuracy of Bayesian inference.

Contribution

It provides a comprehensive review of state-of-the-art post-processing methods for MCMC, including discrepancy minimisation and control variate techniques.

Findings

Discusses limitations of convergence diagnostics

Highlights methods that directly address bias-variance trade-off

Provides guidance on selecting post-processing techniques

Abstract

Markov chain Monte Carlo (MCMC) is the engine of modern Bayesian statistics, being used to approximate the posterior and derived quantities of interest. Despite this, the issue of how the output from a Markov chain is post-processed and reported is often overlooked. Convergence diagnostics can be used to control bias via burn-in removal, but these do not account for (common) situations where a limited computational budget engenders a bias-variance trade-off. The aim of this article is to review state-of-the-art techniques for post-processing Markov chain output. Our review covers methods based on discrepancy minimisation, which directly address the bias-variance trade-off, as well as general-purpose control variate methods for approximating expected quantities of interest.