Leapfrog Hamiltonian Monte Carlo

Assume the inputs of Hamiltonian Monte Carlo and fix a current position $q$, a step size $ϵ>0$, and a finite integer $L\ge 1$.

1. Independently refresh $p\sim N_d(0,M)$ and set $z=(q,p)$.
2. Compute $z_L={\mathrm{\Phi }}_{ϵ}^L(z)$ by the finite recursion in One-step leapfrog map as force and drift shears, then set $z^{\ast }=R(z_L)$, where $R(q,p)=(q,-p)$.
3. Form the deterministic proposal kernel $Q_T(z,A)={\mathbf{1}}_A(z^{\ast })$ on phase space and apply Metropolis–Hastings kernel with target $\nu (dq,dp)=\pi (dq)N_d(0,M)(dp)$. Equivalently, accept $z^{\ast }$ with probability $\min \{1,\exp (-H(z^{\ast })+H(z))\}$ and otherwise retain $z$.
4. Return the position coordinate of the accepted or retained phase point.

The finite recursion is defined for every $(q,p)\in {\mathbb{R}}^{2d}$. No exact ODE solution, energy conservation by the discrete map, or convergence guarantee is part of this procedure.