From Metropolis to diffusions: Gibbs states and optimal scaling.

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From Metropolis to diffusions: Gibbs states and optimal scaling. / Breyer, L. A.; Roberts, G. O.
In: Stochastic Processes and their Applications, Vol. 90, No. 2, 2000, p. 181-206.

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

Harvard

Breyer, LA & Roberts, GO 2000, 'From Metropolis to diffusions: Gibbs states and optimal scaling.', Stochastic Processes and their Applications, vol. 90, no. 2, pp. 181-206. https://doi.org/10.1016/S0304-4149(00)00041-7

APA

Breyer, L. A., & Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Processes and their Applications, 90(2), 181-206. https://doi.org/10.1016/S0304-4149(00)00041-7

Vancouver

Breyer LA, Roberts GO. From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Processes and their Applications. 2000;90(2):181-206. doi: 10.1016/S0304-4149(00)00041-7

Author

Breyer, L. A. ; Roberts, G. O. / From Metropolis to diffusions: Gibbs states and optimal scaling. In: Stochastic Processes and their Applications. 2000 ; Vol. 90, No. 2. pp. 181-206.

Bibtex

@article{810cd1f7869c4291a0e005bfe0975933,

title = "From Metropolis to diffusions: Gibbs states and optimal scaling.",

abstract = "This paper investigates the behaviour of the random walk Metropolis algorithm in high-dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition-free case). Moreover, the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite-dimensional diffusion process in high dimensions.",

keywords = "Markov chain Monte Carlo, Hamiltonians, Hybrid algorithms",

author = "Breyer, {L. A.} and Roberts, {G. O.}",

year = "2000",

doi = "10.1016/S0304-4149(00)00041-7",

language = "English",

volume = "90",

pages = "181--206",

journal = "Stochastic Processes and their Applications",

publisher = "Elsevier",

number = "2",

}

RIS

TY - JOUR

T1 - From Metropolis to diffusions: Gibbs states and optimal scaling.

AU - Breyer, L. A.

AU - Roberts, G. O.

PY - 2000

Y1 - 2000

N2 - This paper investigates the behaviour of the random walk Metropolis algorithm in high-dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition-free case). Moreover, the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite-dimensional diffusion process in high dimensions.

AB - This paper investigates the behaviour of the random walk Metropolis algorithm in high-dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition-free case). Moreover, the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite-dimensional diffusion process in high dimensions.

KW - Markov chain Monte Carlo

KW - Hamiltonians

KW - Hybrid algorithms

U2 - 10.1016/S0304-4149(00)00041-7

DO - 10.1016/S0304-4149(00)00041-7

M3 - Journal article

VL - 90

SP - 181

EP - 206

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 2

ER -

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