About the lower bounds for the multiple testing problem

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.