Metropolis--Hastings with Scalable Subsampling

The Metropolis-Hastings (MH) algorithm is one of the most widely used Markov Chain Monte Carlo schemes for generating samples from Bayesian posterior distributions. The algorithm is asymptotically exact, flexible and easy to implement. However, in the context of Bayesian inference for large datasets, evaluating the likelihood on the full data for thousands of iterations until convergence can be prohibitively expensive. This paper introduces a new subsample MH algorithm that satisfies detailed balance with respect to the target posterior and utilises control variates to enable exact, efficient Bayesian inference on datasets with large numbers of observations. Through theoretical results, simulation experiments and real-world applications on certain generalised linear models, we demonstrate that our method requires substantially smaller subsamples and is computationally more efficient than the standard MH algorithm and other exact subsample MH algorithms.

💡 Research Summary

The paper addresses a fundamental scalability bottleneck of the Metropolis‑Hastings (MH) algorithm in Bayesian inference for massive datasets. Standard MH requires evaluating the full log‑likelihood at every iteration, which becomes prohibitive when the number of observations n reaches millions or billions. Existing remedies—data partitioning, variational approximations, or subsampling‑based MCMC—either sacrifice exactness, struggle with non‑Gaussian posteriors, or suffer from high variance that forces large subsample sizes.

The authors propose a novel “Metropolis‑Hastings with Scalable Subsampling” (MH‑SS) algorithm that retains exactness (the invariant distribution is the true posterior) while dramatically reducing per‑iteration cost. The key technical device is the use of control variates derived from first‑ or second‑order Taylor expansions of each individual log‑likelihood term ℓ_i(θ) around a reference point b_θ (typically an approximation to the posterior mode). For any current state θ and proposal θ′, the exact log‑likelihood difference Σ_i