• Bernoulli
  • Volume 24, Number 4A (2018), 2610-2639.

Perturbation theory for Markov chains via Wasserstein distance

Daniel Rudolf and Nikolaus Schweizer

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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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Bernoulli, Volume 24, Number 4A (2018), 2610-2639.

Received: October 2015
Revised: January 2017
First available in Project Euclid: 26 March 2018

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big data Markov chains MCMC perturbations Wasserstein distance


Rudolf, Daniel; Schweizer, Nikolaus. Perturbation theory for Markov chains via Wasserstein distance. Bernoulli 24 (2018), no. 4A, 2610--2639. doi:10.3150/17-BEJ938.

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