Treffer: A historical perspective on Schützenberger-Pinsker inequalities (extended version)

This paper presents a tutorial overview of so-called Pinsker inequalities which establisha precise relationship between information and statistics, and whose use have becomeubiquitous in many applications. According to Stigler’s law of eponymy, no scientificdiscovery is named after its original discoverer. Pinsker’s inequality is no exception:Years before the publication of Pinsker’s book in 1960, the French medical doctor,geneticist, epidemiologist, and mathematician Marcel-Paul (Marco) Schützenberger,in his 1953 doctoral thesis, not only proved what is now called Pinsker’s inequality(with the optimal constant that Pinsker himself did not establish) but also the optimalsecond-order improvement, more than a decade before Kullback’s derivation of thesame inequality. We review Schützenberger and Pinsker contributions as well as thoseof Volkonskii and Rozanov, Sakaguchi, McKean, Csiszár, Kullback, Kemperman,Vajda, Bretagnolle and Huber, Krafft and Schmitz, Toussaint, Reid and Williamson,Gilardoni, as well as the optimal derivation of Fedotov, Harremoës, and Topsøe. Wealso present some historical elements on the life and work of Schützenberger, anddiscuss an interesting problem of an erroneous constant in the Schützenberger-Pinskerinequality.