HMC: avoiding rejections by not using leapfrog and an analysis of the acceptance rate
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We give numerical evidence that the standard leapfrog algorithm may not be the best integrator to use within the Hamiltonian Monte Carlo (HMC) method and its variants. If the dimensionality of the target distribution is high, the number of accepted proposals obtained with a given computational effort may be multiplied by a factor of three or more by switching to alternative integrators very similar to leapfrog. Such integrators have appeared in the numerical analysis and molecular dynamics literature rather than in the statistics literature. In addition, we provide several theoretical results on the acceptance rate of HMC. We prove that, at stationarity, one out of two accepted proposals comes from an integration leg where energy is lost. We provide a complete study of the acceptance rate in the simplest model problem given by a univariate Gaussian target. A central limit theorem shows that, in a general scenario and for high-dimensional multivariate Gaussian targets, the expected acceptance rate and the expected energy error are related by an equation that does not change with the target, the integrator, the step-length or the number of time-steps. Experiments show that the relation also holds for non-Gaussian targets.
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