Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

Felipe Medina-Aguayo; Daniel Rudolf; Nikolaus Schweizer

doi:10.1016/j.spa.2019.06.015

Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

Stoch Process Their Appl. 2020 Apr;130(4):2200-2227. doi: 10.1016/j.spa.2019.06.015.

Authors

Felipe Medina-Aguayo¹, Daniel Rudolf², Nikolaus Schweizer³

Affiliations

¹ Department of Mathematics and Statistics, University of Reading Whiteknights, PO Box 220, Reading RG6 6AX, United Kingdom.
² Institute for Mathematical Stochastics, Universität Göttingen & Felix-Bernstein-Institute for Mathematical Statistics, Goldschmidtstraße 3-5, 37077 Göttingen, Germany.
³ Department of Econometrics and OR, Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands.

Abstract

The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis-Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the $n$ th step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.

Keywords: Intractable likelihood; Markov chain Monte Carlo; Monte Carlo within Metropolis; Restricted approximation.