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Approximations of geometrically ergodic reversible markov chains

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  22 November 2021

Jeffrey Negrea Open the ORCID record for Jeffrey Negrea [Opens in a new window]  and
Jeffrey S. Rosenthal Open the ORCID record for Jeffrey S. Rosenthal [Opens in a new window]
Jeffrey Negrea*
Affiliation:
University of Toronto
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
*Postal address: Department of Statistical Sciences, 9th Floor, 700 University Ave., Toronto, ON M5G 1Z5, Canada.
*Postal address: Department of Statistical Sciences, 9th Floor, 700 University Ave., Toronto, ON M5G 1Z5, Canada.
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Abstract

A common tool in the practice of Markov chain Monte Carlo (MCMC) is to use approximating transition kernels to speed up computation when the desired kernel is slow to evaluate or is intractable. A limited set of quantitative tools exists to assess the relative accuracy and efficiency of such approximations. We derive a set of tools for such analysis based on the Hilbert space generated by the stationary distribution we intend to sample, $L_2(\pi)$. Our results apply to approximations of reversible chains which are geometrically ergodic, as is typically the case for applications to MCMC. The focus of our work is on determining whether the approximating kernel will preserve the geometric ergodicity of the exact chain, and whether the approximating stationary distribution will be close to the original stationary distribution. For reversible chains, our results extend the results of Johndrow et al. (2015) from the uniformly ergodic case to the geometrically ergodic case, under some additional regularity conditions. We then apply our results to a number of approximate MCMC algorithms.

Keywords

Markov chainMonte Carloperturbations

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J22: Computational methods in Markov chains 65C05: Monte Carlo methods
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 981 - 1022
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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