On Markov chain Monte Carlo methods for tall data

TL;DR

This paper reviews existing scalable MCMC methods for large datasets, proposes a new subsampling approach with theoretical guarantees, and discusses its effectiveness under certain statistical assumptions, highlighting open challenges in less ideal scenarios.

Contribution

The paper provides a comprehensive review of scalable MCMC methods and introduces a novel subsampling-based approach with provable closeness to the posterior distribution under favorable conditions.

Findings

Proposed a subsampling method requiring less than O(n) likelihood evaluations in certain models.

Method performs well when the Bernstein-von Mises approximation is accurate.

Identified open challenges for scenarios with poor Bernstein-von Mises approximation.

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 which samples 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 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.