Open Access
August 2018 On the Poisson equation for Metropolis–Hastings chains
Aleksandar Mijatović, Jure Vogrinc
Bernoulli 24(3): 2401-2428 (August 2018). DOI: 10.3150/17-BEJ932


This paper defines an approximation scheme for a solution of the Poisson equation of a geometrically ergodic Metropolis–Hastings chain $\Phi$. The scheme is based on the idea of weak approximation and gives rise to a natural sequence of control variates for the ergodic average $S_{k}(F)=(1/k)\sum_{i=1}^{k}F(\Phi_{i})$, where $F$ is the force function in the Poisson equation. The main results show that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and give a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.


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Aleksandar Mijatović. Jure Vogrinc. "On the Poisson equation for Metropolis–Hastings chains." Bernoulli 24 (3) 2401 - 2428, August 2018.


Received: 1 August 2016; Revised: 1 October 2016; Published: August 2018
First available in Project Euclid: 2 February 2018

zbMATH: 06839270
MathSciNet: MR3757533
Digital Object Identifier: 10.3150/17-BEJ932

Keywords: asymptotic variance , central limit theorem , Markov chain Monte Carlo , Metropolis–Hastings algorithm , Poisson equation for Markov chains , variance reduction , weak approximation

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 3 • August 2018
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