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The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

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Bierkens, Joris, Fearnhead, Paul and Roberts, Gareth O. (2019) The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. Annals of statistics, 47 (3). pp. 1288-1320. doi:10.1214/18-AOS1715 ISSN 0090-5364.

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Official URL: https://doi.org/10.1214/18-AOS1715

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Abstract

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multi-dimensional version of the Zig-Zag process of Bierkens and Roberts (2017), a continuous time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible non-reversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, i.e. the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial pre-processing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH): Big data, Bayesian statistical decision theory
Journal or Publication Title: Annals of statistics
Publisher: Inst Mathematical Statistics
ISSN: 0090-5364
Official Date: 13 February 2019
Dates:
DateEvent
13 February 2019Published
20 April 2018Accepted
Volume: 47
Number: 3
Page Range: pp. 1288-1320
DOI: 10.1214/18-AOS1715
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): "The right to place the final version of this article (exactly as published in the journal) on their own homepage or in a public digital repository, provided there is a link to the official journal site."
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 24 May 2018
Date of first compliant Open Access: 18 April 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/D002060/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
EP/K014463/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
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