Abstract
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 to obtain approximately independent samples, the computational effort for negative log-density and first coordinate differ: for these BPS requires computational effort whereas ZZ requires . 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).
Citation
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
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