A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

Joris Bierkens; Gareth Roberts

doi:10.1214/16-AAP1217

A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

Joris Bierkens, Gareth Roberts

Statistics

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

47 Citations (Scopus)

79 Downloads (Pure)

Abstract

In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n^1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n^3/4 for LMH and n^3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.

Original language	English
Pages (from-to)	846-882
Number of pages	37
Journal	Annals of Applied Probability
Volume	27
Issue number	2
DOIs	https://doi.org/10.1214/16-AAP1217
Publication status	Published - 2017

Keywords

Exponential ergodicity
Markov chain Monte Carlo
Phase transition
Piecewise deterministic Markov process
Weak convergence

Access to Document

10.1214/16-AAP1217

AAP1217Final published version, 374 KB

Cite this

@article{ecfd395935ff415eb43c605610d219a5,

title = "A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model",

abstract = "In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n3/4 for LMH and n3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic {"}zig-zag{"} Markov process.",

keywords = "Exponential ergodicity, Markov chain Monte Carlo, Phase transition, Piecewise deterministic Markov process, Weak convergence",

author = "Joris Bierkens and Gareth Roberts",

year = "2017",

doi = "10.1214/16-AAP1217",

language = "English",

volume = "27",

pages = "846--882",

journal = "Annals of Applied Probability",

issn = "1050-5164",

publisher = "Institute of Mathematical Statistics",

number = "2",

}

TY - JOUR

T1 - A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

AU - Bierkens, Joris

AU - Roberts, Gareth

PY - 2017

Y1 - 2017

N2 - In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n3/4 for LMH and n3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.

AB - In Turitsyn, Chertkov and Vucelja [Phys. D 240 (2011) 410-414] a nonreversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals n1/2 for LMH, which should be compared to n for MH. At the critical temperature, the required jump rate equals n3/4 for LMH and n3/2 for MH, in agreement with experimental results of Turitsyn, Chertkov and Vucelja (2011). The scaling limit of LMH turns out to be a nonreversible piecewise deterministic exponentially ergodic "zig-zag" Markov process.

KW - Exponential ergodicity

KW - Markov chain Monte Carlo

KW - Phase transition

KW - Piecewise deterministic Markov process

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85019686754&partnerID=8YFLogxK

U2 - 10.1214/16-AAP1217

DO - 10.1214/16-AAP1217

M3 - Article

AN - SCOPUS:85019686754

SN - 1050-5164

VL - 27

SP - 846

EP - 882

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 2

ER -

A piecewise deterministic scaling limit of lifted Metropolis-Hastings in the Curie-Weiss model

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this