The Spectral Representations Of The Simple Hypothesis Testing Problem

TL;DR

This paper derives a spectral duality framework for hypothesis testing error probabilities, providing new bounds and characterizations that extend to general measure spaces and utilize spectral identities and Berry--Esseen theorem.

Contribution

It introduces the primitive entropy spectrum as the convex conjugate of Type II error volume, extending spectral methods to hypothesis testing with randomized detectors.

Findings

Derived the convex conjugate of Type II error probability as an integral of likelihood ratio distribution.

Extended spectral characterization to $\sigma$-finite measures in hypothesis testing.

Provided state-of-the-art bounds for product measures using Berry--Esseen theorem.

Abstract

The convex conjugate (i.e., the Legendre transform) of Type II error probability (volume) as a function of Type I error probability (volume) is determined for the hypothesis testing problem with randomized detectors. The derivation relies on properties of likelihood ratio quantiles and is general enough to extend to the case of $\sigma$-finite measures in all non-trivial cases. The convex conjugate of the Type II error volume, called the primitive entropy spectrum, is expressed as an integral of the complementary distribution function of the likelihood ratio using a standard spectral identity. The resulting dual characterization of the Type II error volume leads to state of the art bounds for the case of product measures via Berry--Esseen theorem through a brief analysis relying on properties of the Gaussian Mills ratio, both with and without tilting.