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Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

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  • Medina-Aguayo, Felipe
  • Rudolf, Daniel
  • Schweizer, Nikolaus

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 nth 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.

Suggested Citation

  • Medina-Aguayo, Felipe & Rudolf, Daniel & Schweizer, Nikolaus, 2020. "Perturbation bounds for Monte Carlo within Metropolis via restricted approximations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2200-2227.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2200-2227
    DOI: 10.1016/j.spa.2019.06.015
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    References listed on IDEAS

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    1. Breyer, Laird & Roberts, Gareth O. & Rosenthal, Jeffrey S., 2001. "A note on geometric ergodicity and floating-point roundoff error," Statistics & Probability Letters, Elsevier, vol. 53(2), pages 123-127, June.
    2. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    3. Jaewoo Park & Murali Haran, 2018. "Bayesian Inference in the Presence of Intractable Normalizing Functions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1372-1390, July.
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    1. Murray Pollock & Paul Fearnhead & Adam M. Johansen & Gareth O. Roberts, 2020. "Quasi‐stationary Monte Carlo and the ScaLE algorithm," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(5), pages 1167-1221, December.

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