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

This paper presents a tutorial overview of so-called Pinsker inequalities which establish a precise relationship between information and statistics, and whose use have become ubiquitous in many applications. According to Stigler’s law of eponymy, no scientific discovery 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 optimal second-order improvement, more than a decade before Kullback’s derivation of the same inequality. We review Schützenberger and Pinsker contributions as well as those of 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. We also present some historical elements on the life and work of Schützenberger, and discuss an interesting problem of an erroneous constant in the Schützenberger-Pinsker inequality.