Discontinuous Hamiltonian Monte Carlo for discrete parameters and discontinuous likelihoods

Hamiltonian Monte Carlo (HMC) is a powerful sampling algorithm employed by several probabilistic programming languages. Its fully automatic implementations have made HMC a standard tool for applied Bayesian modeling. While its performance is often superior to alternatives under a wide range of models, one prominent weakness of HMC is its inability to handle discrete parameters. In this article, we present \textit{discontinuous HMC}, an extension that can efficiently explore discrete spaces involving ordinal parameters as well as target distributions with discontinuous densities. The proposed algorithm is based on two key ideas: embedding of discrete parameters into a continuous space and simulation of Hamiltonian dynamics on a piecewise smooth density function. When properly-tuned, discontinuous HMC is guaranteed to outperform a Metropolis-within-Gibbs algorithm as the two algorithms coincide under a specific (and sub-optimal) implementation of discontinuous HMC. It is additionally shown that the dynamics underlying discontinuous HMC have a remarkable similarity to a zig-zag process, a continuous-time Markov process behind a state-of-the-art non-reversible rejection-free sampler. We apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance.