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A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces

George Wynne, Mikołaj Kasprzak, Andrew B. Duncan

TL;DR

This work extends kernel Stein discrepancy (KSD) to probability measures on separable Hilbert spaces, enabling goodness-of-fit testing for infinite-dimensional objects such as functional data. It introduces a novel Fourier representation that decouples the kernel from the Stein operator, providing conditions under which KSD separates measures and connecting to the Stein–Tikhomirov framework in the Gaussian base case. The paper formulates KSD for Gibbs measures via the generator method, derives an actionable expectation (and hence $U$-statistic) form, and proves a Fourier-based separation result for translation-invariant kernels. Numerical experiments on synthetic functional data validate the framework and illustrate how the Fourier representation clarifies kernel and operator choices, with practical one-sample tests for Gibbs measures and potential extensions to complex infinite-dimensional models.

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.

A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces

TL;DR

This work extends kernel Stein discrepancy (KSD) to probability measures on separable Hilbert spaces, enabling goodness-of-fit testing for infinite-dimensional objects such as functional data. It introduces a novel Fourier representation that decouples the kernel from the Stein operator, providing conditions under which KSD separates measures and connecting to the Stein–Tikhomirov framework in the Gaussian base case. The paper formulates KSD for Gibbs measures via the generator method, derives an actionable expectation (and hence -statistic) form, and proves a Fourier-based separation result for translation-invariant kernels. Numerical experiments on synthetic functional data validate the framework and illustrate how the Fourier representation clarifies kernel and operator choices, with practical one-sample tests for Gibbs measures and potential extensions to complex infinite-dimensional models.

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.
Paper Structure (26 sections, 17 theorems, 103 equations, 2 figures, 3 tables)

This paper contains 26 sections, 17 theorems, 103 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stein's lemma and Stein discrepancies
  4. Kernels and reproducing kernel Hilbert spaces
  5. Probability measures on Hilbert spaces
  6. Derivatives of Hilbert space valued functions
  7. Kernel Stein discrepancy for Gibbs measures
  8. Formulating KSD using the generator method
  9. Expectation formulation of KSD
  10. Fourier representation of kernel Stein discrepancy
  11. Numerical simulations
  12. Testing methodology
  13. Synthetic data goodness-of-fit experiments
  14. Conclusion
  15. Appendix
  16. ...and 11 more sections

Key Result

Proposition 3.1

Suppose that Assumptions ass:X and ass:P_Q_assumptions hold for $Q=P\propto\exp(-U)N_C$, then there exists a $\mathcal{X}$-valued Wiener process $B$ with covariance operator $C$ such that the stochastic differential equation admits a weak solution $(X_t)_{t\geq 0}$ which is a $P$-symmetric diffusion process with invariant measure $P$.

Figures (2)

  • Figure 1: Plots corresponding to Example \ref{['exp:Test_Functions']} of the real part of the test functions $\mathcal{A}(e^{is\cdot})(x)$ for $10$ samples from different choices of $\mu$, the heavier the tails of $\mu$ the larger the samples of $s$ hence the greater the magnitude and periodicity of the test functions. In black is $(\log p)'(x)$ where $p(x)\propto \exp(-\left(\frac{x-3}{3}\right)^{2})$.
  • Figure 2: A plot of KSD using the IMQ kernel against the number of steps in the Euler-Maruyama simulation to simulate the target measure. The target measure is the conditioned SDE \ref{['eq:non_lin_SDE']}. The KSD value was estimated using $2000$ samples of the Euler-Maruyama simulation, keeping the trajectories with $\abs{X(50)}<0.1$.

Theorems & Definitions (39)

  • Example 2.1: Langevin-Stein Operator
  • Example 2.2: Vectorised Langevin-Stein Operator
  • Example 2.3
  • Definition 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.1
  • Proposition 3.2
  • Example 3.1
  • ...and 29 more