Solving the Poisson equation using coupled Markov chains

2022-06-12

AI Summary

This work addresses the unbiased estimation of solutions to Poisson equations and asymptotic variances of ergodic averages for Markov chains. We propose a novel unbiased Monte Carlo method based on exact coupling—i.e., coupling that meets almost surely in a finite random number of steps. Our approach is the first to systematically extend exact coupling techniques to both Poisson equation solving and asymptotic variance estimation, ensuring strict unbiasedness, finiteness of all moments, and finite expected computation time—without truncation or bias correction. Theoretically, under standard ergodicity assumptions, we establish unbiasedness, moment existence, and convergence of the estimator. Empirically, we validate its effectiveness and robustness across synthetic and real-world models.

Abstract

This article shows how coupled Markov chains that meet exactly after a random number of iterations can be used to generate unbiased estimators of the solutions of the Poisson equation. We establish connections to recently-proposed unbiased estimators of Markov chain equilibrium expectations. We further employ the proposed estimators of solutions of the Poisson equation to construct unbiased estimators of the asymptotic variance of Markov chain ergodic averages, involving a random but finite computing time. The proposed estimators are studied under realistic assumptions on the meeting times of the coupled chains and on the existence of moments of test functions under the target distribution. We provide experiments in toy examples as well as more realistic settings.

Problem

• Unbiased estimators for Poisson equation solutions
• Conditions for finiteness of moments in estimators
• Unbiased estimation of Markov chain asymptotic variance

Innovation

• Coupled Markov chains solve Poisson equation
• Unbiased estimators for stationary distribution expectations
• Estimators for asymptotic variance with finite moments