Page - x - in Differential Geometrical Theory of Statistics

x In 1943, Maurice Fréchet wrote a seminal paper (developing elements of his Winter 1939 Lecture at Institut Henri Poincaré in Paris) [23,24] introducing what was then called the Cramer-Rao bound. This paper contains in fact much more than this important discovery. In particular, Maurice Fréchet introduces more general notions relative to "distinguished functions", densities with estimator reaching the bound, defined with a function, solution of Clairaut’s equation. The solutions “envelope of the Clairaut’s equation” are related to standard Legendre transform and basic structures of Information Geometry. This Fréchet’s analysis can also be revisited on the basis of Jean-Louis Koszul works as seminal foundation of “Information Geometry” based on Legendre- Clairaut equation. We can also make references to De Moivre and Leibnitz contributions in seminal development of Probability [25–27] and give reference to papers written on History of probability [28–32]. We thank all the contributors of this edited book for further pushing the envelope of the geometrization of statistics in novel directions. This edited book is organized in six chapters as follows: Chapter I: Geometric Thermodynamics of Jean-Marie Souriau This first chapter introduces and develops Jean-Marie Souriau’s (1922-2012) model of Lie group thermodynamics and relativistic thermodynamics of continua. The contributions are listed below: • From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics by Charles-Michel Marle • Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families by Frédéric Barbaresco • Link between Lie Group Statistical Mechanics and Thermodynamics of Continua by Géry de Saxcé Chapter II: Koszul-Vinberg Model of Hessian Information Geometry The second chapter deals with Jean-Louis Koszul’s model of Hessian Information Geometry based on Koszul-Vinberg’s characteristic function and the homology theory of Koszul-Vinberg algebroids and their modules (KV homology). The two contributions are: • Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology (IN MEMORIAM OF ALEXANDER GROTHENDIECK) by Michel Nguiffo Boyom • Explicit Formula of Koszul–Vinberg Characteristic Functions for a Wide Class of Regular Convex Cones by Hideyuki Ishi Chapter III: Divergence Geometry and Information Geometry The third chapter develops new algorithms related to the area of divergence geometry (minimum divergence estimator, Rényi divergence) and Information Geometry: Mixture of densities, Expectations on q-Exponential Family, Sparse Goodness-of-Fit Testing. The five contributions are: • A Proximal Point Algorithm for Minimum Divergence Estimators with Application to Mixture Models by Diaa Al Mohamad and Michel Broniatowski • Geometry Induced by a Generalization of Rényi Divergence by David C. de Souza, Rui F. Vigelis and Charles C. Cavalcante