Sequential Kernelized Stein Discrepancy

TL;DR

This paper introduces a sequential goodness-of-fit test based on kernelized Stein discrepancy that allows for adaptive stopping and continuous monitoring without requiring boundedness assumptions, demonstrating strong theoretical guarantees and empirical effectiveness.

Contribution

It develops a novel sequential test for unnormalized densities using Stein kernels without boundedness constraints, with proven validity and practical demonstration.

Findings

Test controls false discovery rate during continuous monitoring

The method performs well on various distributions including Boltzmann machines

Provides asymptotic lower bounds for the test's wealth process growth

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.