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Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

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Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. / Neal, Peter; Roberts, Gareth; Yuen, Wai Kong.
In: Annals of Applied Probability, Vol. 22, No. 5, 2012, p. 1880-1927.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Neal, P, Roberts, G & Yuen, WK 2012, 'Optimal scaling of random walk Metropolis algorithms with discontinuous target densities', Annals of Applied Probability, vol. 22, no. 5, pp. 1880-1927. https://doi.org/10.1214/11-AAP817

APA

Neal, P., Roberts, G., & Yuen, W. K. (2012). Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Annals of Applied Probability, 22(5), 1880-1927. https://doi.org/10.1214/11-AAP817

Vancouver

Neal P, Roberts G, Yuen WK. Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Annals of Applied Probability. 2012;22(5):1880-1927. doi: 10.1214/11-AAP817

Author

Neal, Peter ; Roberts, Gareth ; Yuen, Wai Kong. / Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. In: Annals of Applied Probability. 2012 ; Vol. 22, No. 5. pp. 1880-1927.

Bibtex

@article{8079bf345b2c47a2a32e413ca5019064,
title = "Optimal scaling of random walk Metropolis algorithms with discontinuous target densities",
abstract = "We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to ∞. In particular, when the proposal variance is scaled by d−2, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e−2 (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.",
keywords = "Random walk Metropolis, Markov chain Monte Carlo , optimal scaling",
author = "Peter Neal and Gareth Roberts and Yuen, {Wai Kong}",
year = "2012",
doi = "10.1214/11-AAP817",
language = "English",
volume = "22",
pages = "1880--1927",
journal = "Annals of Applied Probability",
issn = "1050-5164",
publisher = "Institute of Mathematical Statistics",
number = "5",

}

RIS

TY - JOUR

T1 - Optimal scaling of random walk Metropolis algorithms with discontinuous target densities

AU - Neal, Peter

AU - Roberts, Gareth

AU - Yuen, Wai Kong

PY - 2012

Y1 - 2012

N2 - We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to ∞. In particular, when the proposal variance is scaled by d−2, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e−2 (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.

AB - We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to ∞. In particular, when the proposal variance is scaled by d−2, the sequence of stochastic processes formed by the first component of each Markov chain converges to an appropriate Langevin diffusion process. Therefore optimizing the efficiency of the RWM algorithm is equivalent to maximizing the speed of the limiting diffusion. This leads to an asymptotic optimal acceptance rate of e−2 (=0.1353) under quite general conditions. The results have major practical implications for the implementation of RWM algorithms by highlighting the detrimental effect of choosing RWM algorithms over Metropolis-within-Gibbs algorithms.

KW - Random walk Metropolis

KW - Markov chain Monte Carlo

KW - optimal scaling

U2 - 10.1214/11-AAP817

DO - 10.1214/11-AAP817

M3 - Journal article

VL - 22

SP - 1880

EP - 1927

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 5

ER -