# Geometry and Dynamics for Markov Chain Monte Carlo

## TL;DR

This paper provides an accessible overview of the geometric and dynamical principles underlying Hamiltonian Monte Carlo, bridging the gap between theory and practice for statisticians and machine learners.

## Contribution

It offers a comprehensive, beginner-friendly introduction to the geometric foundations of Hamiltonian Monte Carlo and discusses recent advances in the field.

## Key findings

- Clarifies the geometric tools used in Hamiltonian Monte Carlo
- Highlights recent theoretical developments and their practical relevance
- Bridges the gap between intuition and formal understanding

## Abstract

Markov Chain Monte Carlo methods have revolutionised mathematical computation and enabled statistical inference within many previously intractable models. In this context, Hamiltonian dynamics have been proposed as an efficient way of building chains which can explore probability densities efficiently. The method emerges from physics and geometry and these links have been extensively studied by a series of authors through the last thirty years. However, there is currently a gap between the intuitions and knowledge of users of the methodology and our deep understanding of these theoretical foundations. The aim of this review is to provide a comprehensive introduction to the geometric tools used in Hamiltonian Monte Carlo at a level accessible to statisticians, machine learners and other users of the methodology with only a basic understanding of Monte Carlo methods. This will be complemented with some discussion of the most recent advances in the field which we believe will become increasingly relevant to applied scientists.