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October 2021 Efficiency of delayed-acceptance random walk Metropolis algorithms
Chris Sherlock, Alexandre H. Thiery, Andrew Golightly
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Ann. Statist. 49(5): 2972-2990 (October 2021). DOI: 10.1214/21-AOS2068

Abstract

Delayed-acceptance Metropolis–Hastings and delayed-acceptance pseudo-marginal Metropolis–Hastings algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased stochastic approximation thereof, but a computationally cheap deterministic approximation is available. An initial accept–reject stage uses the cheap approximation for computing the Metropolis–Hastings ratio; proposals which are accepted at this stage are subjected to a further accept–reject step, which corrects for the error in the approximation. Since the expensive posterior, or the approximation thereof, is only evaluated for proposals which are accepted at the first stage, the cost of the algorithm is reduced and larger scalings may be used.

We focus on the random walk Metropolis (RWM) and consider the delayed-acceptance RWM and the delayed-acceptance pseudo-marginal RWM. We provide a framework for incorporating relatively general deterministic approximations into the theoretical analysis of high-dimensional targets. Justified by diffusion-approximation arguments, we derive expressions for the limiting efficiency and acceptance rates in high-dimensional settings. Finally, these theoretical insights are leveraged to formulate practical guidelines for the efficient tuning of the algorithms. The robustness of these guidelines and predicted properties are verified against simulation studies, all of which are strictly outside of the domain of validity of our limit results.

Funding Statement

AHT acknowledges support from the Singapore Ministry of Education Tier 2 (MOE2016-T2-2-135) and a Young Investigator Award Grant (NUSYIA FY16 P16; R-155-000-180-133).

Citation

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Chris Sherlock. Alexandre H. Thiery. Andrew Golightly. "Efficiency of delayed-acceptance random walk Metropolis algorithms." Ann. Statist. 49 (5) 2972 - 2990, October 2021. https://doi.org/10.1214/21-AOS2068

Information

Received: 1 December 2019; Revised: 1 February 2021; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/21-AOS2068

Subjects:
Primary: 60F05 , 65C05 , 65C40

Keywords: delayed-acceptance , diffusion limit , Markov chain Monte Carlo , pseudo-marginal MCMC , Random-walk metropolis

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 5 • October 2021
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