Perturbation Bounds for Monte Carlo within Metropolis via Restricted Approximations

TL;DR

This paper derives explicit bounds on the distributional differences between perturbed Monte Carlo within Metropolis algorithms and standard Metropolis-Hastings, based on novel Markov chain perturbation results.

Contribution

It provides new explicit perturbation bounds for Markov chains applied to MCwM, extending understanding of approximate sampling methods.

Findings

Explicit bounds for distributional differences derived

Perturbation results applicable beyond MCwM context

Conditions for stability and transition probability control established

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.