A Fourier representation of kernel stein discrepancy with application to goodness-of-fit tests for measures on infinite dimensional Hilbert spaces
File(s)
Accepted version
Author(s)
Wynne, George
Kasprzak, Mikołaj J
Duncan, Andrew B
Type
Journal Article
Abstract
Kernel Stein discrepancy (KSD) is a widely used kernel-based measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. However, so far the method has been restricted to finite-dimensional data. We provide the first analysis of KSD in the generality of data lying in a separable Hilbert space, for example functional data. The main result is a novel Fourier representation of KSD obtained by combining the theory of measure equations with kernel methods. This allows us to prove that KSD can separate measures and thus is valid to use in practice. Additionally, our results improve the interpretability of KSD by decoupling the effect of the kernel and Stein operator. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.
Date Issued
2025-05-01
Date Acceptance
2022-07-01
Citation
Bernoulli, 2025, 31 (2), pp.868-893
ISSN
1350-7265
Publisher
Bernoulli Society for Mathematical Statistics and Probability
Start Page
868
End Page
893
Journal / Book Title
Bernoulli
Volume
31
Issue
2
Copyright Statement
Copyright © 2026 Copyright Owner. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Accepted
Date Publish Online
2025-02-11