Why is HMC faster than random-walk Metropolis?

By using gradient information to propose long trajectories that follow the contours of the posterior, HMC produces nearly independent samples with high acceptance, avoiding the slow diffusive exploration of random-walk methods in high dimensions.

What does HMC require that simpler samplers do not?

It requires the gradient of the log-posterior with respect to continuous parameters, which is why it is typically paired with automatic differentiation and cannot directly handle discrete parameters.

Hamiltonian Monte Carlo

Hamiltonian Monte Carlo uses gradients of the log-posterior and simulated physical dynamics to propose distant, high-acceptance moves, enabling efficient sampling in high dimensions.

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Definition

Hamiltonian Monte Carlo is an MCMC method that introduces auxiliary momentum variables, simulates Hamiltonian dynamics using the gradient of the log-posterior to propose a new state, and accepts it with a Metropolis step that corrects for numerical integration error.

Scope

This topic covers the augmentation of the posterior with momentum variables, the leapfrog integration of Hamiltonian dynamics, the Metropolis correction for discretization error, and the No-U-Turn Sampler that automates path-length and step-size tuning.

Core questions

How do momentum variables and Hamiltonian dynamics produce efficient proposals?
What is the leapfrog integrator and why is the Metropolis correction needed?
How does the No-U-Turn Sampler remove the need to hand-tune trajectory length?
Why does HMC scale better than random-walk methods in high dimensions?

Key concepts

momentum variables
leapfrog integrator
Hamiltonian dynamics
step size
trajectory length
No-U-Turn Sampler
gradient of log-posterior

Key theories

Hamiltonian dynamics for sampling: Augmenting the target with Gaussian momentum and following volume-preserving, energy-conserving dynamics lets the sampler traverse the posterior with high acceptance and low correlation between successive states.
No-U-Turn Sampler: NUTS automatically chooses trajectory lengths by extending the path until it begins to double back, and combines this with step-size adaptation to remove most manual tuning.

Clinical relevance

Hamiltonian Monte Carlo, especially via NUTS, is the default sampler in probabilistic programming systems such as Stan and PyMC, making complex hierarchical models fittable in pharmacometrics, ecology, and the physical sciences.

History

Hybrid Monte Carlo was introduced for lattice quantum chromodynamics by Duane and colleagues in 1987; Neal adapted and popularized it for statistics, and Hoffman and Gelman's 2014 No-U-Turn Sampler made it practical for general users, anchoring modern probabilistic programming.

Debates

Sensitivity to geometry and tuning: HMC can struggle with strongly curved or multimodal posteriors and requires gradient information, prompting work on Riemannian-manifold and adaptive variants.

Key figures

Radford Neal
Simon Duane
Matthew Hoffman
Andrew Gelman
Michael Betancourt

Seminal works

neal2011
hoffman2014

Frequently asked questions

Why is HMC faster than random-walk Metropolis?: By using gradient information to propose long trajectories that follow the contours of the posterior, HMC produces nearly independent samples with high acceptance, avoiding the slow diffusive exploration of random-walk methods in high dimensions.
What does HMC require that simpler samplers do not?: It requires the gradient of the log-posterior with respect to continuous parameters, which is why it is typically paired with automatic differentiation and cannot directly handle discrete parameters.