The Annals of Statistics
- Ann. Statist.
- Volume 47, Number 3 (2019), 1288-1320.
The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data
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 multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], 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 nonreversible 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, that is, 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 preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.
Ann. Statist., Volume 47, Number 3 (2019), 1288-1320.
Received: July 2016
Revised: March 2018
First available in Project Euclid: 13 February 2019
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Bierkens, Joris; Fearnhead, Paul; Roberts, Gareth. The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data. Ann. Statist. 47 (2019), no. 3, 1288--1320. doi:10.1214/18-AOS1715. https://projecteuclid.org/euclid.aos/1550026838
- Supplement to “The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data”. Mathematics of the Zig-Zag process, scaling of SGLD, details on the experiments including how to obtain computational bounds.