Solving the Poisson equation using coupled Markov chains

06/12/2022

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by   Randal Douc, et al.

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This article draws connections between unbiased estimators constructed from coupled Markov chains that meet exactly after a random number of iterations, and solutions of the Poisson equation. We first show how such pairs of chains can be employed to obtain unbiased estimators of pointwise evaluations of solutions of the Poisson equation. We then propose new estimators of the asymptotic variance of Markov chain ergodic averages. 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 illustrate their behaviour in toy examples and in a more challenging setting of high-dimensional Bayesian regression.