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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 ISSN 1050-5164.

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Official URL: https://doi.org/10.1214/19-AAP1467

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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
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > 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
Dates:
DateEvent
18 October 2019Published
8 April 2019Accepted
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 (Creative Commons)
Date of first compliant deposit: 8 April 2019
Date of first compliant Open Access: 11 April 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/N000188/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
1501767National Science Foundationhttp://dx.doi.org/10.13039/100000001
DGE-114747National Science Foundationhttp://dx.doi.org/10.13039/100000001
Frederick E. Terman FellowshipStanford Universityhttp://dx.doi.org/10.13039/100005492
UNSPECIFIEDLloyd's Register Foundationhttp://dx.doi.org/10.13039/100008885
UNSPECIFIEDAlan Turing Institutehttp://dx.doi.org/10.13039/100012338
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