Dimension-free Relaxation Times of Informed MCMC Samplers on Discrete Spaces

Hyunwoong Chang; Quan Zhou

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arXiv · 2404.03867

Dimension-free Relaxation Times of Informed MCMC Samplers on Discrete Spaces

Hyunwoong Chang·Quan Zhou

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Abstract

Convergence analysis of Markov chain Monte Carlo methods in high-dimensional statistical applications is increasingly recognized. In this paper, we develop general mixing time bounds for Metropolis-Hastings algorithms on discrete spaces by building upon and refining some recent theoretical advancements in Bayesian model selection problems. We establish sufficient conditions for a class of informed Metropolis-Hastings algorithms to attain relaxation times that are independent of the problem dimension. These conditions are grounded in the high-dimensional statistical theory and allow for possibly multimodal posterior distributions. We obtain our results through two independent techniques: the multicommodity flow method and single-element drift condition analysis; we find that the latter yields a slightly tighter mixing time bound. Our results are readily applicable to a broad spectrum of statistical problems with discrete parameter spaces, as we demonstrate using both theoretical and numerical examples.

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Hyunwoong Chang, Quan Zhou. 2024-04-05. Dimension-free Relaxation Times of Informed MCMC Samplers on Discrete Spaces. https://arxiv.org/abs/2404.03867

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