Metropolis–Hastings transition kernel couplings

John O’Leary; Guanyang Wang

doi:10.1214/22-AIHP1360

Metropolis–Hastings transition kernel couplings

Research output: Contribution to journal › Article › peer-review

Abstract

Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis–Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O’Leary, Wang and Jacob (In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (2021) 1225–1233 PMLR). Our results represent an advance in understanding the MH transition kernel and a step forward for coupling this popular class of algorithms.

Original language	American English
Pages (from-to)	1101-1124
Number of pages	24
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	60
Issue number	2
DOIs	https://doi.org/10.1214/22-AIHP1360
State	Published - May 2024
Externally published	Yes

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Couplings
Markov chain Monte Carlo
Metropolis–Hastings algorithm

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10.1214/22-AIHP1360

Cite this

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abstract = "Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis–Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O{\textquoteright}Leary, Wang and Jacob (In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (2021) 1225–1233 PMLR). Our results represent an advance in understanding the MH transition kernel and a step forward for coupling this popular class of algorithms.",

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AU - O’Leary, John

AU - Wang, Guanyang

N1 - Publisher Copyright: © Association des Publications de l’Institut Henri Poincaré, 2024.

PY - 2024/5

Y1 - 2024/5

N2 - Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis–Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O’Leary, Wang and Jacob (In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (2021) 1225–1233 PMLR). Our results represent an advance in understanding the MH transition kernel and a step forward for coupling this popular class of algorithms.

AB - Couplings play a central role in the analysis of Markov chain convergence and in the construction of novel Markov chain Monte Carlo estimators, diagnostics, and variance reduction techniques. The set of possible couplings is often intractable, frustrating the search for tight bounds and efficient estimators. To address this challenge for algorithms in the Metropolis–Hastings (MH) family, we establish a simple characterization of the set of MH transition kernel couplings. We then extend this result to describe the set of maximal couplings of the MH kernel, resolving an open question of O’Leary, Wang and Jacob (In Proceedings of The 24th International Conference on Artificial Intelligence and Statistics (2021) 1225–1233 PMLR). Our results represent an advance in understanding the MH transition kernel and a step forward for coupling this popular class of algorithms.

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Metropolis–Hastings transition kernel couplings

Abstract

ASJC Scopus subject areas

Keywords

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