A short note on an inequality between KL and TV
arXiv:2202.07198
Abstract
The goal of this short note is to discuss the relation between Kullback--Leibler divergence and total variation distance, starting with the celebrated Pinsker's inequality relating the two, before switching to a simple, yet (arguably) more useful inequality, apparently not as well known, due to Bretagnolle and Huber. We also discuss applications of this bound for minimax testing lower bounds.
Update: positive answer to Question 8 ("from the TFL to the BH bound"), communicated to me by Hao-Chung Cheng