Reflection, refraction, and Hamiltonian Monte Carlo

Hamiltonian Monte Carlo (HMC) is a successful approach for sampling from continuous densities. However, it has difficulty simulating Hamiltonian dynamics with non-smooth functions, leading to poor performance. This paper is motivated by the behavior of Hamiltonian dynamics in physical systems like optics. We introduce a modification of the Leapfrog discretization of Hamiltonian dynamics on piecewise continuous energies, where intersections of the trajectory with discontinuities are detected, and the momentum is reflected or refracted to compensate for the change in energy. We prove that this method preserves the correct stationary distribution when boundaries are affine. Experiments show that by reducing the number of rejected samples, this method improves on traditional HMC.