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The zig-zag and super-efficient sampling for Bayesian analysis of big data

Presented by: 
Gareth Roberts
Monday 15th January 2018 - 16:10 to 16:55
INI Seminar Room 1
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. The talk will discuss a new family of Monte Carlo methods based upon a multi-dimensional version of the Zig-Zag process of (Bierkens, Roberts, 2016), a continuous time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction the Zig-Zag process offers a flexible non-reversible alternative. The dynamics of the Zig-Zag process correspond to a constant velocity model, with the velocity of the process switching at events from a point process. The rate of this point process can be related to the invariant distribution of the process. If we wish to target a given posterior distribution, then rates need to be set equal to the gradient of the log of the posterior. Unlike traditional MCMC, Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, I will discuss two generalisations which have good scaling properties for big data: firstly a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme; and secondly a control-variate variant of the sub-sampling idea to reduce the variance of our unbiased estimator. Very recent ergodic theory will also be described.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons