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Optimal design of the Barker proposal and other locally-balanced Metropolis-Hastings algorithms

Vogrinc, Jure; Livingstone, Samuel; Zanella, Giacomo; (2022) Optimal design of the Barker proposal and other locally-balanced Metropolis-Hastings algorithms. Biometrika (In press). Green open access

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Abstract

We study the class of first-order locally balanced Metropolis–Hastings algorithms introduced in Livingstone & Zanella (2022). To choose a specific algorithm within the class, the user must select a balancing function g:R+→R+ satisfying g(t)=tg(1/t) and a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a general limiting optimal acceptance rate of 57% and scaling of n−1/3 ⁠, as the dimension n tends to infinity among all members of the class under mild smoothness assumptions on g and when the target distribution for the algorithm is of product form. In particular, we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimize this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, an optimal choice of balancing function under a Gaussian noise distribution, and an optimal choice of first-order locally balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings, and in particular, show that a bimodal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.

Type: Article
Title: Optimal design of the Barker proposal and other locally-balanced Metropolis-Hastings algorithms
Open access status: An open access version is available from UCL Discovery
Publisher version: https://academic.oup.com/biomet
Language: English
Additional information: Copyright 2022 Biometrika Trust, This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
UCL classification: UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL
URI: https://discovery.ucl.ac.uk/id/eprint/10156310
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