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December 2020 Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo
Alain Durmus, Éric Moulines, Eero Saksman
Ann. Statist. 48(6): 3545-3564 (December 2020). DOI: 10.1214/19-AOS1941

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.

Citation

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Alain Durmus. Éric Moulines. Eero Saksman. "Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo." Ann. Statist. 48 (6) 3545 - 3564, December 2020. https://doi.org/10.1214/19-AOS1941

Information

Received: 1 May 2019; Revised: 1 December 2019; Published: December 2020
First available in Project Euclid: 11 December 2020

MathSciNet: MR4185819
Digital Object Identifier: 10.1214/19-AOS1941

Subjects:
Primary: 60J05, 60J22, 65C05

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 6 • December 2020
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