Solving the Poisson equation using coupled Markov
This work starts with connections between couplings of Markov chains and solutions of the Poisson equation. We show how pairs of chains can be employed to obtain unbiased estimators of pointwise evaluations of solutions of the Poisson equation. Motivated by MCMC, we propose new estimators of the asymptotic variance of Markov chain ergodic averages. The proposed estimators have distinct appeals and drawbacks relative to standard methods, such as batch means or spectral variance methods. We formally study the proposed estimators 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 describe experiments in toy examples and more challenging settings arising in Bayesian analysis. This is joint work with Randal Douc, Anthony Lee and Dootika Vats.