Variational formulas for the power of the binary hypothesis testing problem with applications

TL;DR

This paper introduces two variational formulas for the power of binary hypothesis testing, linking them to the Legendre transform and the CDF of the log-likelihood ratio, with applications to bounds and asymptotic analysis.

Contribution

It derives novel variational formulas for hypothesis testing power, connecting them to divergence measures and providing a framework for asymptotic and non-asymptotic analysis.

Findings

Upper bound on power using Rényi divergence

Framework for asymptotic and non-asymptotic power expressions

Demonstration in central limit and large deviations regimes

Abstract

Two variational formulas for the power of the binary hypothesis testing problem are derived. The first is given as the Legendre transform of a certain function and the second, induced from the first, is given in terms of the Cumulative Distribution Function (CDF) of the log-likelihood ratio. One application of the first formula is an upper bound on the power of the binary hypothesis testing problem in terms of the Re'nyi divergence. The second formula provide a general framework for proving asymptotic and non-asymptotic expressions for the power of the test utilizing corresponding expressions for the CDF of the log-likelihood. The framework is demonstrated in the central limit regime (i.e., for non-vanishing type I error) and in the large deviations regime.