Noisy Hamiltonian Monte Carlo for Doubly Intractable Distributions

ABSTRACT

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician’s toolbox as an alternative sampling method in settings when standard Metropolis–Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept–reject step is used to correct the bias. For doubly intractable distributions—such as posterior distributions based on Gibbs random fields—HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept–reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.

KEYWORDS:

• Bayesian inference
• Hamiltonian Monte Carlo
• Intractable likelihood
• Markov chain Monte Carlo
• Markov random fields

Acknowledgments

The authors thank the anonymous referees and the editors whose valuable comments and insightful suggestions led to an improved version of the article.

Funding

The Insight Centre for Data Analytics is supported by Science Foundation Ireland under Grant Number SFI/12/RC/2289. Alan Benson and Nial Friel’s research was also supported by a Science Foundation Ireland grant: 12/IP/1424.