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

Perturbation Bounds for Monte Carlo within Metropolis via Restricted Approximations