Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

Radu V. Craiu; Lawrence Gray; Krzysztof Łatuszyński; Neal Madras; Gareth O. Roberts; Jeffrey S. Rosenthal

doi:10.1214/14-AAP1083

Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

Radu V. Craiu, Lawrence Gray, Krzysztof Łatuszyński, Neal Madras, Gareth O. Roberts, Jeffrey S. Rosenthal

Mathematics

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10 Scopus citations

Abstract

We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.

Original language	English (US)
Pages (from-to)	3592-3623
Number of pages	32
Journal	Annals of Applied Probability
Volume	25
Issue number	6
DOIs	https://doi.org/10.1214/14-AAP1083
State	Published - Dec 1 2015

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Publisher Copyright:
© 2015 Institute of Mathematical Statistics.

Keywords

Adaptive MCMC algorithms
Convergence
Ergodicity
Markov chain
Perturbation
Stability

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AU - Łatuszyński, Krzysztof

AU - Madras, Neal

AU - Roberts, Gareth O.

AU - Rosenthal, Jeffrey S.

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AB - We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms.

KW - Adaptive MCMC algorithms

KW - Convergence

KW - Ergodicity

KW - Markov chain

KW - Perturbation

KW - Stability

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Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms

Abstract

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Keywords

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