Hamiltonian-Assisted Metropolis Sampling

Abstract

Various Markov chain Monte Carlo (MCMC) methods are studied to improve upon random walk Metropolis sampling, for simulation from complex distributions. Examples include Metropolis-adjusted Langevin algorithms, Hamiltonian Monte Carlo, and other algorithms related to underdamped Langevin dynamics. We propose a broad class of irreversible sampling algorithms, called Hamiltonian-assisted Metropolis sampling (HAMS), and develop two specific algorithms with appropriate tuning and preconditioning strategies. Our HAMS algorithms are designed to simultaneously achieve two distinctive properties, while using an augmented target density with a momentum as an auxiliary variable. One is generalized detailed balance, which induces an irreversible exploration of the target. The other is a rejection-free property for a Gaussian target with a prespecified variance matrix. This property allows our preconditioned algorithms to perform satisfactorily with relatively large step sizes. Furthermore, we formulate a framework of generalized Metropolis–Hastings sampling, which not only highlights our construction of HAMS at a more abstract level, but also facilitates possible further development of irreversible MCMC algorithms. We present several numerical experiments, where the proposed algorithms consistently yield superior results among existing algorithms using the same preconditioning schemes.

KEYWORDS:

• Auxiliary variables
• Detailed balance
• Hamiltonian Monte Carlo
• Markov chain Monte Carlo
• Metropolis-adjusted Langevin algorithms
• Metropolis–Hastings sampling
• Underdamped Langevin dynamics

Acknowledgments

The authors acknowledge the Office of Advanced Research Computing at Rutgers University for providing access to computing resources for the numerical studies reported here. The authors also thank two referees for helpful comments.

Supplementary Material

Appendices:

(I) Auxiliary variable derivation, (II) Validity of UDL, (III) Generalized Metropolis-Hastings sampling, (IV) Proofs, (V) Details for simulation studies, (VI) Additional simulation results. (pdf).

Computer codes:

R and Python codes for simulation studies in Section 5 and Supplement Section VI. (zip).