The Physicist's Guide to the HMC

TL;DR

This paper reviews recent advancements in the hybrid Monte Carlo (HMC) algorithm, emphasizing its efficiency and improvements like exact Fourier acceleration and radial updates for sampling from complex probability distributions.

Contribution

It compiles recent enhancements of HMC, including EFA and radial updates, into a practical numerical recipe for improved sampling efficiency.

Findings

HMC is highly efficient for continuous distributions.

EFA makes HMC equivalent to direct sampling for quadratic actions.

Recent improvements enhance HMC's applicability to arbitrary actions.

Abstract

The hybrid Monte Carlo (HMC) algorithm is arguably the most efficient sampling method for general probability distributions of continuous variables. Together with exact Fourier acceleration (EFA) the HMC becomes equivalent to direct sampling for quadratic actions $S(x)=\frac12 x^\mathsf{T} M x$ (i.e. normal distributions $x\sim \mathrm{e}^{-S(x)}$), only perturbatively worse for perturbative deviations of the action from the quadratic case, and it remains viable for arbitrary actions. In this work the most recent improvements of the HMC including EFA and radial updates are collected into a numerical recipe.