Merging MCMC Subposteriors through Gaussian-Process Approximations

TL;DR

This paper introduces a novel divide-and-conquer MCMC approach that uses Gaussian-process approximations of subposteriors to efficiently estimate the full Bayesian posterior in large-data settings.

Contribution

It proposes a new method for merging subposteriors via Gaussian-process models, enabling scalable Bayesian inference with improved approximation accuracy.

Findings

The Gaussian-process approximation effectively captures subposterior densities.

The method provides accurate posterior expectation estimates.

It offers a measure of uncertainty in the estimates.

Abstract

Markov chain Monte Carlo (MCMC) algorithms have become powerful tools for Bayesian inference. However, they do not scale well to large-data problems. Divide-and-conquer strategies, which split the data into batches and, for each batch, run independent MCMC algorithms targeting the corresponding subposterior, can spread the computational burden across a number of separate workers. The challenge with such strategies is in recombining the subposteriors to approximate the full posterior. By creating a Gaussian-process approximation for each log-subposterior density we create a tractable approximation for the full posterior. This approximation is exploited through three methodologies: firstly a Hamiltonian Monte Carlo algorithm targeting the expectation of the posterior density provides a sample from an approximation to the posterior; secondly, evaluating the true posterior at the sampled points leads to an importance sampler that, asymptotically, targets the true posterior expectations; finally, an alternative importance sampler uses the full Gaussian-process distribution of the approximation to the log-posterior density to re-weight any initial sample and provide both an estimate of the posterior expectation and a measure of the uncertainty in it.