October 2022 High-dimensional scaling limits of piecewise deterministic sampling algorithms
Joris Bierkens, Kengo Kamatani, Gareth O. Roberts
Author Affiliations +
Ann. Appl. Probab. 32(5): 3361-3407 (October 2022). DOI: 10.1214/21-AAP1762


Piecewise deterministic Markov processes are an important new tool in the design of Markov chain Monte Carlo algorithms. Two examples of fundamental importance are the bouncy particle sampler (BPS) and the zig–zag process (ZZ). In this paper scaling limits for both algorithms are determined. Here the dimensionality of the space tends towards infinity and the target distribution is the multivariate standard normal distribution. For several quantities of interest (angular momentum, first coordinate and negative log-density) the scaling limits show qualitatively very different and rich behaviour. Based on these scaling limits the performance of the two algorithms in high dimensions can be compared. Although for angular momentum both processes require only a computational effort of O(d) to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires O(d2) computational effort whereas ZZ requires O(d). Finally we provide a criterion for the choice of the refreshment rate of BPS.

Funding Statement

JB is supported by the research programme “Zig-zagging through computational barriers” with project number 016.Vidi.189.043, financed by the Netherlands Organisation for Scientific Research (NWO). KK is supported by JST CREST Grant Number JPMJCR14D7 and JSPS KAKENHI Grant Number JP16K00046. GOR is supported by the EPSRC grants: ilike (EP/K014463/1), CoSInES (EP/R034710/1) and Bayes for Health (EP/R018561/1).


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Joris Bierkens. Kengo Kamatani. Gareth O. Roberts. "High-dimensional scaling limits of piecewise deterministic sampling algorithms." Ann. Appl. Probab. 32 (5) 3361 - 3407, October 2022. https://doi.org/10.1214/21-AAP1762


Received: 1 July 2019; Revised: 1 May 2021; Published: October 2022
First available in Project Euclid: 18 October 2022

MathSciNet: MR4497848
zbMATH: 1504.65002
Digital Object Identifier: 10.1214/21-AAP1762

Primary: 60F05
Secondary: 65C05

Keywords: exponential ergodicity , Gaussian process , Markov chain Monte Carlo , Piecewise deterministic Markov processes , weak convergence

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 5 • October 2022
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