 Hamiltonian Monte Carlo

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.