Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods.

Hamiltonian Monte Carlo has emerged as a standard tool for posterior computation. In this article, we present an extension that can efficiently explore target distributions with discontinuous densities, which in turn enables efficient sampling from ordinal parameters though embedding of probability mass functions into continuous spaces. We motivate our approach through a theory of discontinuous Hamiltonian dynamics and develop a numerical solver of discontinuous dynamics. The proposed numerical solver is the first of its kind, with a remarkable ability to exactly preserve the Hamiltonian and thus yield a type of rejection-free proposals. We apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance.