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Optimal tuning of the hybrid monte-carlo algorithm

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Beskos, Alexandros, Pillai, Natesh S., Roberts, Gareth O., Sanz-Serna, Jesus and Stuart, A. M. (2013) Optimal tuning of the hybrid monte-carlo algorithm. Bernoulli, Volume 19 (Number 5a). pp. 1501-1534. doi:10.3150/12-BEJ414 ISSN 1350-7265.

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Official URL: http://dx.doi.org/10.3150/12-BEJ414

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Abstract

We investigate the properties of the Hybrid Monte Carlo algorithm (HMC) in high dimensions.
HMC develops a Markov chain reversible w.r.t. a given target distribution . by using separable Hamiltonian dynamics with potential -log .. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-
Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an O(1) acceptance probability as the dimension d of the state space tends to ., the leapfrog step-size h should be scaled as h=l Ɨdāˆ’1/
4 . Therefore, in high dimensions, HMC requires O(d1/
4 ) steps to traverse the state space. We also identify
analytically the asymptotically optimal acceptance probability, which turns out to be 0.651
(to three decimal places). This is the choice which optimally balances the cost of generating a
proposal, which decreases as l increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as l increases

Item Type: Journal Article
Alternative Title:
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Faculty of Science, Engineering and Medicine > Science > Statistics
Journal or Publication Title: Bernoulli
Publisher: Int Statistical Institute
ISSN: 1350-7265
Official Date: 2013
Dates:
DateEvent
2013Published
Volume: Volume 19
Number: Number 5a
Page Range: pp. 1501-1534
DOI: 10.3150/12-BEJ414
Status: Peer Reviewed
Publication Status: Published

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