Open Access
October 2019 Measuring sample quality with diffusions
Jackson Gorham, Andrew B. Duncan, Sebastian J. Vollmer, Lester Mackey
Ann. Appl. Probab. 29(5): 2884-2928 (October 2019). DOI: 10.1214/19-AAP1467


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 diffusion. 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 hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship 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.


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Jackson Gorham. Andrew B. Duncan. Sebastian J. Vollmer. Lester Mackey. "Measuring sample quality with diffusions." Ann. Appl. Probab. 29 (5) 2884 - 2928, October 2019.


Received: 1 February 2018; Revised: 1 November 2018; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155062
MathSciNet: MR4019878
Digital Object Identifier: 10.1214/19-AAP1467

Primary: 60E15 , 60J60 , 62-04 , 62E17 , 65C60
Secondary: 62-07 , 65C05 , 68T05

Keywords: Itô diffusion , Markov chain Monte Carlo , Multivariate Stein factors , sample quality , Stein discrepancy , Stein’s method , Wasserstein decay

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.29 • No. 5 • October 2019
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