The block-Poisson estimator for optimally tuned exact subsampling MCMC

TL;DR

This paper introduces a novel block-Poisson estimator for pseudo-marginal MCMC that enables efficient likelihood estimation through data subsampling, achieving high correlation between likelihood estimates and improving sampling efficiency.

Contribution

The paper proposes a new block-Poisson estimator for subsampling MCMC that allows for correlated likelihood estimates, reducing the required subsample size and improving efficiency.

Findings

The method achieves higher efficiency compared to standard MCMC without subsampling.

It outperforms two recent exact subsampling approaches in the literature.

Guidelines for optimal parameter tuning are provided.

Abstract

Speeding up Markov Chain Monte Carlo (MCMC) for datasets with many observations by data subsampling has recently received considerable attention. A pseudo-marginal MCMC method is proposed that estimates the likelihood by data subsampling using a block-Poisson estimator. The estimator is a product of Poisson estimators, allowing us to update a single block of subsample indicators in each MCMC iteration so that a desired correlation is achieved between the logs of successive likelihood estimates. This is important since pseudo-marginal MCMC with positively correlated likelihood estimates can use substantially smaller subsamples without adversely affecting the sampling efficiency. The block-Poisson estimator is unbiased but not necessarily positive, so the algorithm runs the MCMC on the absolute value of the likelihood estimator and uses an importance sampling correction to obtain consistent estimates of the posterior mean of any function of the parameters. Our article derives guidelines to select the optimal tuning parameters for our method and shows that it compares very favourably to regular MCMC without subsampling, and to two other recently proposed exact subsampling approaches in the literature.