Zig-zag sampling for discrete structures and non-reversible phylogenetic MCMC

We construct a zig-zag process targeting a posterior distribution defined on a hybrid state space consisting of both discrete and continuous variables. The construction does not require any assumptions on the structure among discrete variables. We demonstrate our method on two examples in genetics based on the Kingman coalescent, showing that the zig-zag process can lead to efficiency gains of up to several orders of magnitude over classical Metropolis--Hastings algorithms, and that it is well suited to parallel computation. Our construction resembles existing techniques for Hamiltonian Monte Carlo on a hybrid state space, which suffers from implementationally and analytically complex boundary crossings when applied to the coalescent. We demonstrate that the continuous-time zig-zag process avoids these complications.