On Markov chain Monte Carlo methods for tall data
Rémi Bardenet, Arnaud Doucet, Chris Holmes.
Year: 2017, Volume: 18, Issue: 47, Pages: 1−43
Abstract
Markov chain Monte Carlo methods are often deemed too computationally intensive to be of any practical use for big data applications, and in particular for inference on datasets containing a large number $n$ of individual data points, also known as tall datasets. In scenarios where data are assumed independent, various approaches to scale up the Metropolis- Hastings algorithm in a Bayesian inference context have been recently proposed in machine learning and computational statistics. These approaches can be grouped into two categories: divide-and-conquer approaches and, subsampling-based algorithms. The aims of this article are as follows. First, we present a comprehensive review of the existing literature, commenting on the underlying assumptions and theoretical guarantees of each method. Second, by leveraging our understanding of these limitations, we propose an original subsampling-based approach relying on a control variate method which samples under regularity conditions from a distribution provably close to the posterior distribution of interest, yet can require less than $O(n)$ data point likelihood evaluations at each iteration for certain statistical models in favourable scenarios. Finally, we emphasize that we have only been able so far to propose subsampling-based methods which display good performance in scenarios where the Bernstein-von Mises approximation of the target posterior distribution is excellent. It remains an open challenge to develop such methods in scenarios where the Bernstein-von Mises approximation is poor.