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Measuring sample quality with diffusions
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Gorham, Jackson, Duncan, Andrew B., Vollmer, Sebastian and Mackey, Lester (2019) Measuring sample quality with diffusions. Annals of Applied Probability, 29 (5). pp. 2884-2928. doi:10.1214/19-AAP1467
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Official URL: https://doi.org/10.1214/19-AAP1467
Abstract
Stein’s method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Itô diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Itô diffusions. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyper parameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules, and quantify bias-variance trade-offs in approximate MCMC. Our results establish a near-linear relation-ship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest
| Item Type: | Journal Article | |||||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||
| Divisions: | Faculty of Science > Mathematics | |||||||||||||||||||||
| Library of Congress Subject Headings (LCSH): | Distribution (Probability theory), Markov processes, Monte Carlo method | |||||||||||||||||||||
| Journal or Publication Title: | Annals of Applied Probability | |||||||||||||||||||||
| Publisher: | Institute of Mathematical Statistics | |||||||||||||||||||||
| ISSN: | 1050-5164 | |||||||||||||||||||||
| Official Date: | 18 October 2019 | |||||||||||||||||||||
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| Volume: | 29 | |||||||||||||||||||||
| Number: | 5 | |||||||||||||||||||||
| Page Range: | pp. 2884-2928 | |||||||||||||||||||||
| DOI: | 10.1214/19-AAP1467 | |||||||||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||||||||
| Publication Status: | Published | |||||||||||||||||||||
| Access rights to Published version: | Open Access | |||||||||||||||||||||
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