Hamiltonian Monte Carlo ($H(q,p)=-\log \pi (q)+p^T{M}^{-1}p/2$)

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.

Surrogate Hamiltonian Monte Carlo uses an auxiliary smooth probability density function $\nu$ to generate trajectories but applies a Metropolis–Hastings acceptance probability using the intended target $\mu$. For Gaussian momenta and an involutive Metropolis proposal generated by flipped leapfrog integration, the acceptance probability is $\min (1,\mu (x^{\prime })e^{-\Vert p^{\prime }{\Vert }^2/2}/[\mu (x)e^{-\Vert p{\Vert }^2/2}])$. Only the surrogate gradient is required during the trajectory; target density evaluations are still required at the endpoints.