  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. Our extension in
particular 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
corresponding numerical solver. The proposed solver is the first of its kind,
with a remarkable ability to exactly preserve the Hamiltonian. We apply our
algorithm to challenging posterior inference problems to demonstrate its wide
applicability and competitive performance.
