On the Poisson equation for Metropolis-Hastings chains

Aleksandar Mijatovic; Jure Vogrinc

On the Poisson equation for Metropolis-Hastings chains

Probability 2017-02-28 v2 Methodology

Authors: Aleksandar Mijatovic , 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 approximations give 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 result of the paper shows that the sequence of the asymptotic variances (in the CLTs for the control-variate estimators) converges to zero and gives a rate of this convergence. Numerical examples in the case of a double-well potential are discussed.

Keywords

Poisson equation Metropolis-Hastings chain geometric ergodicity control variates ergodic average asymptotic variance central limit theorem

Cite

@article{arxiv.1511.07464,
  title  = {On the Poisson equation for Metropolis-Hastings chains},
  author = {Aleksandar Mijatovic and Jure Vogrinc},
  journal= {arXiv preprint arXiv:1511.07464},
  year   = {2017}
}

Comments

Presentation streamlined, new short proof of Proposition 3.2 in the reversible case with other arguments essentially unchanged, 25 pages, no figures, to appear in Bernoulli

On the Poisson equation for Metropolis-Hastings chains

Abstract

Keywords

Cite

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