Sequential Kernelized Stein Discrepancy
Diego Martinez-Taboada, Aaditya Ramdas
TL;DR
Sequential Kernelized Stein Discrepancy introduces a sequential, anytime-valid goodness-of-fit test for unnormalized densities by marrying kernelized Stein discrepancy with a betting-based martingale framework. It removes the common requirement of uniform Stein kernel boundedness by exploiting pointwise bounds and a normalization $M_p$ to construct nonnegative wealth processes, enabling continuous monitoring and adaptive stopping. The paper proves validity under the null and establishes exponential wealth growth under alternatives, with empirical demonstrations on Gaussian, intractable, and Gaussian-Bernoulli RBM models. This approach broadens KS-based goodness-of-fit testing to complex energy-based models and MCMC diagnostics, offering resource-efficient, flexible hypothesis testing in practice.
Abstract
We present a sequential version of the kernelized Stein discrepancy goodness-of-fit test, which allows for conducting goodness-of-fit tests for unnormalized densities that are continuously monitored and adaptively stopped. That is, the sample size need not be fixed prior to data collection; the practitioner can choose whether to stop the test or continue to gather evidence at any time while controlling the false discovery rate. In stark contrast to related literature, we do not impose uniform boundedness on the Stein kernel. Instead, we exploit the potential boundedness of the Stein kernel at arbitrary point evaluations to define test martingales, that give way to the subsequent novel sequential tests. We prove the validity of the test, as well as an asymptotic lower bound for the logarithmic growth of the wealth process under the alternative. We further illustrate the empirical performance of the test with a variety of distributions, including restricted Boltzmann machines.