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On the accept-reject mechanism for Metropolis-Hastings algorithms
Journal article   Open access   Peer reviewed

On the accept-reject mechanism for Metropolis-Hastings algorithms

Nathan Glatt-holtz, Justin Krometis and Cecilia Mondaini
The Annals of applied probability, v 33(6B), pp 5279-5333
01 Dec 2023
DOI: https://doi.org/10.1214/23-AAP1948
url
https://arxiv.org/pdf/2011.04493View
Submitted Open arXiv.org - Non-exclusive license to distribute

Abstract

Science & Technology Statistics & Probability Mathematics Physical Sciences
This work develops a powerful and versatile framework for determin-ing 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 meth-ods with more complicated "extended phase space" algorithms based around ideas from Hamiltonian dynamics. Our starting point is an abstract result de-veloped in the generality of measurable state spaces that addresses proposal kernels that possess a certain involution structure. Note that, while this under-lying 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 ab-solutely 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 gen-eral 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 precondi-tioned Crank-Nicolson (pCN), infinity MALA and infinity 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.

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