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.

翻译：本短文的目的是讨论Kullback-Leibler散度与全变差距离之间的关系。首先回顾联系这两者的著名Pinsker不等式，然后转向一个形式简单但（可以说）更有用的不等式——该不等式由Bretagnolle和Huber提出，但似乎并不广为人知。我们还讨论了这一界限在极小极大检验下界中的应用。