Page - ix - in Differential Geometrical Theory of Statistics

ix objective of this edited book is to come back to this initial idea that we can geometrize statistics in a rigorous way. We can also make reference to Blaise Pascal for this book on computing geometrical statistics, because he was the inventor of the computer with his “Pascaline” machine. The introduction of Pascaline marks the beginning of the development of mechanical calculus in Europe. This development, which will traverse from the calculating machines to the electrical and electronic calculators of the following centuries, will culminate with the invention of the microprocessor. However, it was also Charles Babbage who conceived his analytical machine from 1834 to 1837, a programmable calculating machine which was the ancestor of the computers of the 1940s, combining the inventions of Blaise Pascal and Jacquard’s machine, with instructions written on perforated cards. One of the descendants of the Pascaline, this was the first machine which performed with the intelligence of man. Figure 2. La « pascaline », Computing Machine, Blaise Pascal 1645 Before introducing the chapters of this book, let us recall that the modern birth of information geometry in the 20th century started with the differential-geometric modeling of parametric family of distributions in the pioneer work of Professor Harold Hotelling in 1929 and in Prodessor Maurice Fréchet Lecture at IHP (Institut Henri Poincaré, Paris) during Winter 1939. Professor Hotelling spent half a year collaborating with Sir Ronald A. Fisher on setting the firm foundation of mathematical statistics in Rothamsted Research (UK) [20–22]. He submitted a groundbreaking note entitled “Spaces of Statistical Parameters” to the American Mathematical Society (AMS) meeting in 1929. Since he did not join the meeting, the note was nevertheless read by Prof. O. Ore. In this work, he introduced the Fisher information metric and the induced Riemannian geometry for modeling parametric family of distributions. C. R. Rao later independently introduced this geometric structure in his celebrated paper entitled “Information and the accuracy attainable in the estimation of statistical parameters” (1945). This paper is truly exceptional since it introduces three key results: (1) Cramér-Rao lower bound, (2) Riemannian geometry of statistical spaces, and (3) Rao-Blackwellization of estimators.