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Perturbation analysis of Markov chain Monte Carlo for graphical models

Part of: Markov processes

Published online by Cambridge University Press:  06 January 2025

Na Lin ,
Yuanyuan Liu  and
Aaron Smith
Na Lin*
Affiliation:
Central South University
Yuanyuan Liu*
Affiliation:
Central South University
Aaron Smith*
Affiliation:
University of Ottawa
*
*Postal address: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, China.
*Postal address: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, China.
***Postal address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada.
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Abstract

The basic question in perturbation analysis of Markov chains is: how do small changes in the transition kernels of Markov chains translate to chains in their stationary distributions? Many papers on the subject have shown, roughly, that the change in stationary distribution is small as long as the change in the kernel is much less than some measure of the convergence rate. This result is essentially sharp for generic Markov chains. We show that much larger errors, up to size roughly the square root of the convergence rate, are permissible for many target distributions associated with graphical models. The main motivation for this work comes from computational statistics, where there is often a tradeoff between the per-step error and per-step cost of approximate MCMC algorithms. Our results show that larger perturbations (and thus less-expensive chains) still give results with small error.

Keywords

Perturbation errormixing timeapproximate MCMCHellinger distancedecay of correlations

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 22
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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