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November 2018 Perturbation theory for Markov chains via Wasserstein distance
Daniel Rudolf, Nikolaus Schweizer
Bernoulli 24(4A): 2610-2639 (November 2018). DOI: 10.3150/17-BEJ938

Abstract

Perturbation theory for Markov chains addresses the question of how small differences in the transition probabilities of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the $n$th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis–Hastings and stochastic Langevin algorithms.

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Daniel Rudolf. Nikolaus Schweizer. "Perturbation theory for Markov chains via Wasserstein distance." Bernoulli 24 (4A) 2610 - 2639, November 2018. https://doi.org/10.3150/17-BEJ938

Information

Received: 1 October 2015; Revised: 1 January 2017; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853259
MathSciNet: MR3779696
Digital Object Identifier: 10.3150/17-BEJ938

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4A • November 2018
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