MCMC using Hamiltonian dynamics

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.

💡 Research Summary

The paper provides a thorough review of Hamiltonian Monte Carlo (HMC), a Markov‑chain Monte Carlo method that leverages Hamiltonian dynamics to generate distant, high‑probability proposals for the Metropolis algorithm. Traditional random‑walk Metropolis schemes suffer from diffusive behavior, especially in high‑dimensional continuous spaces, leading to slow exploration and high autocorrelation. HMC overcomes this by augmenting the target variable (\theta) with an auxiliary momentum variable (p) drawn from a Gaussian distribution. The joint density is expressed as (\pi(\theta,p)\propto\exp