Hamiltonian Monte Carlo for (Physics) Dummies

2026-01-04

AI Summary

This study addresses the challenge faced by non-physics researchers in understanding Hamiltonian Monte Carlo (HMC) by offering an application-oriented, systematic tutorial review. It provides an intuitive exposition of HMC’s sampling mechanism grounded in Hamiltonian dynamics, bridging theoretical foundations with practical implementation in Bayesian inference. The work clarifies how HMC achieves efficient and accurate posterior sampling in complex models and surveys its prominent variants in statistical computing and machine learning. Its central contribution lies in substantially lowering the conceptual barrier to HMC, thereby narrowing the gap between theoretical underpinnings and black-box software tools, and enhancing the algorithm’s accessibility, transparency, and principled use within the broader research community.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Sampling-based inference has seen a surge of interest in recent years. Hamiltonian Monte Carlo (HMC) has emerged as a powerful algorithm that leverages concepts from Hamiltonian dynamics to efficiently explore complex target distributions. Variants of HMC are available in popular software packages, enabling off-the-shelf implementations that have greatly benefited the statistics and machine learning communities. At the same time, the availability of such black-box implementations has made it challenging for users to understand the inner workings of HMC, especially when they are unfamiliar with the underlying physical principles. We provide a pedagogical overview of HMC that aims to bridge the gap between its theoretical foundations and practical applicability. This review article seeks to make HMC more accessible to applied researchers by highlighting its advantages, limitations, and role in enabling scalable and exact Bayesian inference for complex models.

Problem

• Hamiltonian Monte Carlo
• sampling-based inference
• Bayesian inference
• Hamiltonian dynamics
• black-box implementations

Innovation

• Hamiltonian Monte Carlo
• Bayesian inference
• pedagogical review
• statistical computing
• scalable inference