About the lower bounds for the multiple testing problem

TL;DR

This paper provides a concise proof of key lower bounds in the multiple testing problem, improving existing inequalities and unifying several fundamental results in statistical hypothesis testing.

Contribution

It offers a simplified, unified proof of lower bounds like Fano's lemma and enhances the inequality by Venkataramanan and Johnson, advancing theoretical understanding.

Findings

Unified proof of multiple testing lower bounds

Improved inequality for error probability bounds

Simplification of existing theoretical results

Abstract

Given an observed random variable, consider the problem of recovering its distribution among a family of candidate ones. The two-point inequality, Fano's lemma and more recently an inequality due to Venkataramanan and Johnson (2018) allow to bound the maximal probability of error over the family from below. The aim of this paper is to give a very short and simple proof of all these results simultaneously and improve in passing the inequality of Venkataramanan and Johnson.