MCMC using Hamiltonian dynamics

TL;DR

Hamiltonian Monte Carlo leverages Hamiltonian dynamics to efficiently explore continuous state spaces in Markov Chain Monte Carlo methods, improving sampling speed and accuracy by preserving volume and enabling complex proposals.

Contribution

This paper reviews theoretical foundations and practical variations of Hamiltonian Monte Carlo, highlighting new techniques for trajectory computation, acceptance decisions, and mode handling.

Findings

Hamiltonian dynamics enable distant proposals, reducing random walk behavior.

Variations like windowed acceptance and tempering improve sampling efficiency.

Approximate trajectory computation accelerates the method without losing volume preservation.

Abstract

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though originating in physics, Hamiltonian dynamics can be applied to most problems with continuous state spaces by simply introducing fictitious "momentum" variables. A key to its usefulness is that Hamiltonian dynamics preserves volume, and its trajectories can thus be used to define complex mappings without the need to account for a hard-to-compute Jacobian factor - a property that can be exactly maintained even when the dynamics is approximated by discretizing time. In this review, I discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.