A Conceptual Introduction to Hamiltonian Monte Carlo

TL;DR

This paper offers a comprehensive, accessible conceptual overview of Hamiltonian Monte Carlo, explaining its theoretical foundations, practical applications, and limitations to both practitioners and statisticians.

Contribution

It provides a clear, intuitive understanding of Hamiltonian Monte Carlo's principles, bridging the gap between complex mathematics and practical implementation.

Findings

Clarifies the conditions for HMC's success

Explains when HMC is likely to fail

Provides guidance on optimal HMC implementation

Abstract

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.