The Annals of Applied Probability

Measuring sample quality with diffusions

Jackson Gorham, Andrew B. Duncan, Sebastian J. Vollmer, and Lester Mackey

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 5 (2019), 2884-2928.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60J60: Diffusion processes [See also 58J65] 62-04: Explicit machine computation and programs (not the theory of computation or programming) 62E17: Approximations to distributions (nonasymptotic) 60E15: Inequalities; stochastic orderings 65C60: Computational problems in statistics
Secondary: 62-07: Data analysis 65C05: Monte Carlo methods 68T05: Learning and adaptive systems [See also 68Q32, 91E40]

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


Gorham, Jackson; Duncan, Andrew B.; Vollmer, Sebastian J.; Mackey, Lester. Measuring sample quality with diffusions. Ann. Appl. Probab. 29 (2019), no. 5, 2884--2928. doi:10.1214/19-AAP1467.

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