Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo

Abstract

Hamiltonian Monte Carlo (HMC) is currently one of the most popular Markov Chain Monte Carlo algorithms to sample smooth distributions over continuous state space. This paper discusses the irreducibility and geometric ergodicity of the HMC algorithm. We consider cases where the number of steps of the Störmer–Verlet integrator is either fixed or random. Under mild conditions on the potential $U$ associated with target distribution $\pi$, we first show that the Markov kernel associated to the HMC algorithm is irreducible and positive recurrent. Under more stringent conditions, we then establish that the Markov kernel is Harris recurrent. We provide verifiable conditions on $U$ under which the HMC sampler is geometrically ergodic. Finally, we illustrate our results on several examples.