A splitting method to reduce MCMC variance

TL;DR

This paper investigates splitting and killing methods to enhance MCMC accuracy for rare event probability estimation, proving the optimality of weighted ensemble and demonstrating significant variance reduction in practice.

Contribution

It proves weighted ensemble is uniquely asymptotically consistent with MCMC, establishes a lower variance bound, and shows practical variance reduction through numerical examples.

Findings

Weighted ensemble is the only asymptotically consistent splitting method with MCMC.

A lower bound on the asymptotic variance of weighted ensemble estimates.

Weighted ensemble can reduce MCMC variance by multiple orders of magnitude.

Abstract

We explore whether splitting and killing methods can improve the accuracy of Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we make three contributions. First, we prove that "weighted ensemble" is the only splitting and killing method that provides asymptotically consistent estimates when combined with MCMC. Second, we prove a lower bound on the asymptotic variance of weighted ensemble's estimates. Third, we give a constructive proof and numerical examples to show that weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude.