Limit theorems for the zig-zag process
OA Location
Author(s)
Bierkens, Joris
Duncan, Andrew
Type
Journal Article
Abstract
Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.
Date Issued
2017-09-01
Date Acceptance
2017-09-01
Citation
Advances in Applied Probability, 2017, 49 (3), pp.791-825
ISSN
0001-8678
Publisher
Applied Probability Trust
Start Page
791
End Page
825
Journal / Book Title
Advances in Applied Probability
Volume
49
Issue
3
Copyright Statement
© 2017 Applied Probability Trust. This article has been published in a revised form in Advances in Applied Probability [https://doi.org/10.1017/apr.2017.22]. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works.
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000416417500006&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Grant Number
EP/J009636/1
EP/L020564/1
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
MCMC
nonreversible Markov process
piecewise deterministic Markov process
continuous-time Markov process
central limit theorem
functional central limit theorem
LONG-TIME BEHAVIOR
MONTE-CARLO
VARIANCE REDUCTION
LANGEVIN SAMPLERS
MARKOV-PROCESSES
CHAINS
CONVERGENCE
EQUATION
Publication Status
Published
Date Publish Online
2017-09-08