## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 2401-2428.

### On the Poisson equation for Metropolis–Hastings chains

Aleksandar Mijatović and Jure Vogrinc

#### Abstract

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.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 2401-2428.

**Dates**

Received: August 2016

Revised: October 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540478

**Digital Object Identifier**

doi:10.3150/17-BEJ932

**Mathematical Reviews number (MathSciNet)**

MR3757533

**Zentralblatt MATH identifier**

06839270

**Keywords**

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

#### Citation

Mijatović, Aleksandar; Vogrinc, Jure. On the Poisson equation for Metropolis–Hastings chains. Bernoulli 24 (2018), no. 3, 2401--2428. doi:10.3150/17-BEJ932. https://projecteuclid.org/euclid.bj/1517540478