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Geometric integrators and the Hamiltonian Monte Carlo method

Published online by Cambridge University Press:  04 May 2018

Nawaf Bou-Rabee  and
J. M. Sanz-Serna
Nawaf Bou-Rabee
Affiliation:
Department of Mathematical Sciences, Rutgers University Camden, 311 N. Fifth Street, CamdenNJ 08102, USA E-mail: nawaf.bourabee@rutgers.edu
J. M. Sanz-Serna
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E–28911 Leganés (Madrid), Spain E-mail: jmsanzserna@gmail.com
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Abstract

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.

Type
Research Article
Information
Acta Numerica , Volume 27 , 01 May 2018 , pp. 113 - 206
Copyright
© Cambridge University Press, 2018 

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