HMC: avoiding rejections by not using leapfrog and an analysis of the acceptance rate

12/06/2019
by   M. P. Calvo, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/06/2019

HMC: avoiding rejections by not using leapfrog and some results on the acceptance rate

We give numerical evidence that the standard leapfrog algorithm may not ...
research
06/06/2021

On Irreversible Metropolis Sampling Related to Langevin Dynamics

There has been considerable interest in designing Markov chain Monte Car...
research
11/14/2017

Geometric integrators and the Hamiltonian Monte Carlo method

This paper surveys in detail the relations between numerical integration...
research
06/14/2022

Conservative Hamiltonian Monte Carlo

We introduce a new class of Hamiltonian Monte Carlo (HMC) algorithm call...
research
07/05/2023

Adaptive multi-stage integration schemes for Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a powerful tool for Bayesian statistica...
research
09/10/2022

Parallel MCMC Algorithms: Theoretical Foundations, Algorithm Design, Case Studies

Parallel Markov Chain Monte Carlo (pMCMC) algorithms generate clouds of ...
research
12/12/2022

Acceptance Rates of Invertible Neural Networks on Electron Spectra from Near-Critical Laser-Plasmas: A Comparison

While the interaction of ultra-intense ultra-short laser pulses with nea...

Please sign up or login with your details

Forgot password? Click here to reset