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Letbe the Hamiltonian. The Hamiltonian Monte Carlo proposal is accepted with the Metropolis–Hastings acceptance probabilityThe leapfrog map is volume preserving and reversible after momentum reversal, so no proposal-density Jacobian appears.

Hamiltonian Monte Carlo augments the position $x$ by an independent momentum $p\sim N(0,M)$ and uses the Hamiltonian functionFrom the current $x$, draw a fresh $p$, apply a fixed number of leapfrog steps to approximate Hamiltonian flow, and obtain $(x^{\prime },p^{\prime })$. The Metropolis–Hastings acceptance probability isotherwise retain $x$. Momentum negation may be included to make the proposal explicitly reversible. The leapfrog map is volume preserving and reversible, while the acceptance step corrects its discretization error, leaving $\pi$ invariant after the momentum is discarded.

Radford M. Neal works on Bayesian inference, machine learning and computational methods for complex probability distributions. His research page lists neural networks, Gaussian processes and Markov chain Monte Carlo.^[ref]

His MCMC Using Hamiltonian Dynamics by Radford M. Neal develops Hamiltonian Monte Carlo: introduce auxiliary momenta, simulate approximately energy-preserving motion and correct discretization error with a Metropolis–Hastings algorithm. Longer, directed proposals can explore a target more effectively than a diffusive random walk. The combination of volume preservation, reversibility and an acceptance correction explains why an approximate numerical trajectory can still produce the intended invariant distribution.^[ref]