Hamiltonian Monte Carlo

Assume AC. Let $d\ge 1$, let $U:{\mathbb{R}}^d\to \mathbb{R}$ be continuously differentiable, and suppose $Z_U:={\int }_{{\mathbb{R}}^d}{e}^{-U(q)}dq\in (0,\mathrm{\infty })\mathrm{.}$ The position target is the Borel probability with density $\pi (q)=Z_U^{-1}{e}^{-U(q)}\mathrm{.}$ Let $M$ be a real symmetric positive-definite $d\times d$ mass matrix. Define the kinetic energy and Hamiltonian by $K(p)=\frac{1}{2}{p}^T{M}^{-1}p,H(q,p)=U(q)+K(p)\mathrm{.}$ At each iteration, refresh the momentum independently from $p\sim N_d(0,M)$, the multivariate normal law of mean zero and covariance $M$ (Multivariate normal law, including singular covariance). The corresponding Hamilton equations are $\dot{q}=M^{-1}p,\dot{p}=-\mathrm{\nabla }U(q),$ where $\mathrm{\nabla }U$ is the Euclidean gradient of the objective under the convention of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case and the BCS-20 gradient-descent input contract Gradient descent.

For finite leapfrog integration, choose $ϵ>0$ and an integer $L\ge 1$, use the map from One-step leapfrog map as force and drift shears for $L$ steps, and flip the final momentum. If $z=(q,p)$, write $(\tilde{q},\tilde{p})={\mathrm{\Phi }}_{ϵ}^L(z)$ and define the flipped proposal $z^{\ast }=(\tilde{q},-\tilde{p})$. The Metropolis acceptance probability is $\alpha (z,z^{\ast })=\min \{1,\exp (-H(z^{\ast })+H(z))\}\mathrm{.}$ The general accepted-flow and rejected-self-transition convention is the Metropolis-Hastings procedure Metropolis–Hastings kernel.

The finite leapfrog procedure is defined by finitely many evaluations of $\mathrm{\nabla }U$ and does not require the Hamilton ODE to be complete. An exact-flow version requires $U\in C^2$ and a complete Hamiltonian vector field; under AC, which supplies Countable Choice by AC supplies countable selections and prescribed serial paths, A complete Hamiltonian flow for a C2 potential preserves energy and phase volume proves its energy and volume properties.

Neal's high-dimensional scaling analysis uses a replicated independent-coordinate target of the form $U(q)={\sum }_i{u}_i(q_i)$ (with the coordinate contributions independently replicated). This definition makes no scaling claim outside that regime.