Understanding the Hamiltonian Monte Carlo through its Physics Fundamentals and Examples

TL;DR

This paper explains the physical principles behind Hamiltonian Monte Carlo, providing insights into its mechanics, and compares its performance with other MCMC methods through examples and Python code.

Contribution

It offers a comprehensive explanation of Hamiltonian dynamics in HMC and demonstrates its advantages over other algorithms with practical examples and code.

Findings

HMC outperforms RWMH and t-walk in efficiency for Bayesian inference

Physical understanding enhances the effective application of HMC

Python implementations facilitate practical adoption

Abstract

The Hamiltonian Monte Carlo (HMC) algorithm is a powerful Markov Chain Monte Carlo (MCMC) method that uses Hamiltonian dynamics to generate samples from a target distribution. To fully exploit its potential, we must understand how Hamiltonian dynamics work and why they can be used in a MCMC algorithm. This work elucidates the Monte Carlo Hamiltonian, providing comprehensive explanations of the underlying physical concepts. It is intended for readers with a solid foundation in mathematics who may lack familiarity with specific physical concepts, such as those related to Hamiltonian dynamics. Additionally, we provide Python code for the HMC algorithm, examples and comparisons with the Random Walk Metropolis-Hastings (RWMH) and t-walk algorithms to highlight HMC's strengths and weaknesses when applied to Bayesian Inference.