Abstract
This work develops a powerful and versatile framework for determining acceptance ratios in Metropolis–Hastings-type Markov kernels widely used in statistical sampling problems. Our approach allows us to derive new classes of kernels which unify random walk or diffusion-type sampling methods with more complicated “extended phase space” algorithms based around ideas from Hamiltonian dynamics. Our starting point is an abstract result developed in the generality of measurable state spaces that addresses proposal kernels that possess a certain involution structure. Note that, while this underlying proposal structure suggests a scope which includes Hamiltonian-type kernels, we demonstrate that our abstract result is, in an appropriate sense, equivalent to an earlier general state space setting developed in (Ann. Appl. Probab. 8 (1998) 1–9) where the connection to Hamiltonian methods was more obscure.
On the basis of our abstract results we develop several new classes of extended phase space, HMC-like algorithms. First we tackle the classical finite-dimensional setting of a continuously distributed target measure. We then consider an infinite-dimensional framework for targets which are absolutely continuous with respect to a Gaussian measure with a trace-class covariance. Each of these algorithm classes can be viewed as “surrogate-trajectory” methods, providing a versatile methodology to bypass expensive gradient computations through skillful reduced order modeling and/or data driven approaches as we begin to explore in a forthcoming companion work (Glatt-Holtz et al. (2023)). On the other hand, along with the connection of our main abstract result to the framework in (Ann. Appl. Probab. 8 (1998) 1–9), these algorithm classes provide a unifying picture connecting together a number of popular existing algorithms which arise as special cases of our general frameworks under suitable parameter choices. In particular we show that, in the finite-dimensional setting, we can produce an algorithm class which includes the Metropolis adjusted Langevin algorithm (MALA) and random walk Metropolis method (RWMC) alongside a number of variants of the HMC algorithm including the geometric approach introduced in (J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 (2011) 123–214). In the infinite-dimensional situation, we show that the algorithm class we derive includes the preconditioned Crank–Nicolson (pCN), ∞MALA and ∞HMC methods considered in (Stoch. Dyn. 8 (2008) 319–350; Stochastic Process. Appl. 121 (2011) 2201–2230; Statist. Sci. 28 (2013) 424–446) as special cases.
Funding Statement
Our efforts are supported under the Grants DMS-1816551 and NSF-DMS-2108790 (NEGH, DMS-2009859 (CFM), and NSF-DMS-2108791 (JAK).
Acknowledgments
This work was partially conceived over the course of two research visits: by CFM to New Orleans, LA in January 2020 and by NEGH and CFM to Blacksburg, VA in February 2020. We are grateful to the Tulane and Virginia Tech math departments for hosting these productive visits.
We would like to thank a number of our colleagues for extensive and invaluable feedback surrounding this work: Geordie Richards for additional references concerning recent developments in the study of Gibbs measures for dispersive PDEs, Andrew Holbrook for extensive references from the machine learning and statistics literature and specifically for pointing us to the Metropolis–Hastings–Green algorithm as well as a body of recent work on surrogate trajectory methods, and Gideon Simpson for suggesting the extension of Theorem 2.1 to include a more general proposal space Y in the formulation found in the second draft of this manuscript. Finally we would like to thank Radford Neal, Michael Betancourt, Shiwei Lan and Christophe Andrieu for further inspiring conversations that helped us contextualize this work.
Citation
Nathan Glatt-Holtz. Justin Krometis. Cecilia Mondaini. "On the accept–reject mechanism for Metropolis–Hastings algorithms." Ann. Appl. Probab. 33 (6B) 5279 - 5333, December 2023. https://doi.org/10.1214/23-AAP1948
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