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Hamiltonian Monte Carlo samples a smooth target density $\pi (q)$ by augmenting position $q$ with independent Gaussian momentum $p\sim N(0,M)$, where $M$ is positive-definite. Its joint density is proportional to $e^{-H(q,p)}$, withAfter refreshing momentum, approximately solve Hamilton's equations $\dot{q}=M^{-1}p$ and $\dot{p}=\nabla \log \pi (q)$ using leapfrog integration. Flip momentum to make the fixed-length proposal reversible, then accept with probability $\min (1,e^{H(q,p)-H(q^{\prime },p^{\prime })})$. A unit Jacobian determinant and reversibility justify this Metropolis–Hastings acceptance probability for integration error.^[ref]

Momentum sustains movement through the target instead of proposing each position by a short random increment. The step size, trajectory length and mass matrix affect efficiency; invariance alone does not guarantee ergodicity or efficient crossing between isolated modes.

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]