Split Hamiltonian Monte Carlo revisited

TL;DR

This paper analyzes split Hamiltonian Monte Carlo methods, revealing stability issues and demonstrating that preconditioning can significantly enhance sampler efficiency compared to standard leapfrog HMC.

Contribution

It provides a detailed study of splitting strategies in HMC, showing how preconditioning can overcome stability restrictions and improve sampling efficiency.

Findings

Splitting the Hamiltonian can cause stability issues similar to leapfrog.

Preconditioning the dynamics alleviates stability restrictions.

Combined splitting and preconditioning lead to more efficient samplers.

Abstract

We study Hamiltonian Monte Carlo (HMC) samplers based on splitting the Hamiltonian $H$ as $H_0(\theta,p)+U_1(\theta)$, where $H_0$ is quadratic and $U_1$ small. We show that, in general, such samplers suffer from stepsize stability restrictions similar to those of algorithms based on the standard leapfrog integrator. The restrictions may be circumvented by preconditioning the dynamics. Numerical experiments show that, when the $H_0(\theta,p)+U_1(\theta)$ splitting is combined with preconditioning, it is possible to construct samplers far more efficient than standard leapfrog HMC.