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The Spectral Representations Of The Simple Hypothesis Testing Problem

Barış Nakiboğlu

TL;DR

The work develops a non-asymptotic, spectral theory for simple hypothesis testing with randomized detectors under σ-finite measures. Central to the approach is the primitive entropy spectrum ${h(\cdot|_{{q}}^{{w}})}$ and the LR quantiles, which yield the dual representation ${\beta(\epsilon|w^q)=\sup_{\gamma\ge0} h(\gamma|w^q)-\gamma\epsilon}$ and connect the Type II error volume to the convex analysis framework. A new Change of Measure Lemma enables exact relations between different measure pairs via tilting, broadening Neyman–Pearson results beyond probability measures. The paper then provides sharp, non-asymptotic approximations for ${\beta}$ in memoryless and tilted settings, using Berry–Esseen bounds and Gaussian Mills ratio controls, illustrating the practical power of the spectral representation for finite-length hypothesis testing and potential extensions to broader information-theoretic problems.

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 $σ$-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.

The Spectral Representations Of The Simple Hypothesis Testing Problem

TL;DR

The work develops a non-asymptotic, spectral theory for simple hypothesis testing with randomized detectors under σ-finite measures. Central to the approach is the primitive entropy spectrum and the LR quantiles, which yield the dual representation and connect the Type II error volume to the convex analysis framework. A new Change of Measure Lemma enables exact relations between different measure pairs via tilting, broadening Neyman–Pearson results beyond probability measures. The paper then provides sharp, non-asymptotic approximations for in memoryless and tilted settings, using Berry–Esseen bounds and Gaussian Mills ratio controls, illustrating the practical power of the spectral representation for finite-length hypothesis testing and potential extensions to broader information-theoretic problems.

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 -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.
Paper Structure (19 sections, 10 theorems, 115 equations)

This paper contains 19 sections, 10 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. An Instructive Elementary Decision Problem
  3. Prior Work On $\beta \left({\cdot}\middle\vert_{{\it {w}}}^{{\it {q}}}\!\right)$, $\IfNoValueTF{\vee}{F_{\ell}\!\left({\cdot}\middle\vert_{{\it {q}}}^{{\it {w}}}\!\right)} {F_{\ell\!}^{\vee}\!\left({\cdot}\middle\vert_{{\it {q}}}^{{\it {w}}}\!\right)}$, and ${\mathit{h}} \left({\cdot}\middle\vert_{{\it {w}}}^{{\it {q}}}\!\right)$
  4. Prior Work On Approximations For $\beta$
  5. Main Contributions
  6. Outline
  7. Preliminaries
  8. Notational Conventions
  9. Convex Functions on the Real Line
  10. The Primitive Entropy Spectrum
  11. The Semicontinuous Inverses of Monotonic Functions and The Likelihood Ratio Quantiles
  12. A Spectral Neyman--Pearson Lemma
  13. Likelihood Ratio Tests
  14. The Spectral Representation of $\beta \left({\cdot}\middle\vert_{{\it {w}}}^{{\it {q}}}\!\right)$
  15. A Change of Measure Lemma For $\beta \left({\cdot}\middle\vert_{{\it {w}}}^{{\it {q}}}\!\right)$
  16. ...and 4 more sections

Key Result

Lemma 1

For any $(\!{\it {w}},{\it {q}},{\it {\nu}}\!)$ triple satisfying Assumption assumption:model and eq:assumption:model, i.e., ${\mathit{h}} \left({1}\middle\vert_{{\it {q}}}^{{\it {w}}}\!\right)\!<\!\infty$, Hence, either ${\mathit{h}} \left({\cdot}\middle\vert_{{\it {q}}}^{{\it {w}}}\!\right)\!=\!0$, or ${\mathit{h}} \left({\cdot}\middle\vert_{{\it {q}}}^{{\it

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Remark 1
  • proof : Proof of Lemma \ref{['lem:EntropySpectrum']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:ES']}
  • Definition 3
  • Remark 2
  • Definition 4
  • ...and 19 more