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

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

Bierkens, G.N.J.C. (TU Delft Statistics)
Fearnhead, Paul (Lancaster University)
Roberts, Gareth (University of Warwick)

2019

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.

MCMC
nonreversible Markov process
piecewise deterministic Markov process
stochastic gradient Langevin dynamics
sub-sampling
exact sampling

http://resolver.tudelft.nl/uuid:c8aa0421-e47f-48c3-8611-9bd8a55d6f2c

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

0090-5364

Annals of Statistics, 47 (3), 1288-1320

Institutional Repository

journal article

© 2019 G.N.J.C. Bierkens, Paul Fearnhead, Gareth Roberts