Sub-Gaussian Error Bounds for Hypothesis Testing

TL;DR

This paper introduces a geometric approach to hypothesis testing using KL divergence and sub-Gaussian norms, providing new error bounds that improve upon classical inequalities especially in low-sample or small-hypothesis regimes.

Contribution

It proposes a novel geometric interpretation of likelihood tests and derives new error bounds based on sub-Gaussian norms and KL divergence, enhancing existing bounds like Pinsker's and Fano's.

Findings

Derived sub-Gaussian error bounds for binary hypothesis testing.

Extended bounds to M-ary hypothesis testing, more informative in certain regimes.

Provided a geometric perspective linking KL divergence and sub-Gaussian variables.

Abstract

We interpret likelihood-based test functions from a geometric perspective where the Kullback-Leibler (KL) divergence is adopted to quantify the distance from a distribution to another. Such a test function can be seen as a sub-Gaussian random variable, and we propose a principled way to calculate its corresponding sub-Gaussian norm. Then an error bound for binary hypothesis testing can be obtained in terms of the sub-Gaussian norm and the KL divergence, which is more informative than Pinsker's bound when the significance level is prescribed. For $M$-ary hypothesis testing, we also derive an error bound which is complementary to Fano's inequality by being more informative when the number of hypotheses or the sample size is not large.