Title: Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods,
Abstract: 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.