PDMP Monte Carlo methods for piecewise-smooth densities

There has been substantial interest in developing Markov chain Monte Carlo algorithms based on piecewise deterministic Markov processes. However, existing algorithms can only be used if the target distribution of interest is differentiable everywhere. The key to adapting these algorithms so that they can sample from densities with discontinuities is to define appropriate dynamics for the process when it hits a discontinuity. We present a simple condition for the transition of the process at a discontinuity which can be used to extend any existing sampler for smooth densities, and give specific choices for this transition which work with popular algorithms such as the bouncy particle sampler, the coordinate sampler, and the zigzag process. Our theoretical results extend and make rigorous arguments that have been presented previously, for instance constructing samplers for continuous densities restricted to a bounded domain, and we present a version of the zigzag process that can work in such a scenario. Our novel approach to deriving the invariant distribution of a piecewise deterministic Markov process with boundaries may be of independent interest.