The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data
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Publication:144289
DOI10.1214/18-AOS1715zbMATH Open1417.65008arXiv1607.03188OpenAlexW3104370808WikidataQ128371771 ScholiaQ128371771MaRDI QIDQ144289
Author name not available (Why is that?)
Publication date: 11 July 2016
Published in: (Search for Journal in Brave)
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, 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 {em 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.
Full work available at URL: https://arxiv.org/abs/1607.03188
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