Page - 139 - in Differential Geometrical Theory of Statistics

Article Foliations-Webs-HessianGeometry-Information Geometry-EntropyandCohomology† MichelNguiffoBoyom ALEXANDERGROTHENDIECKINSTITUTE, IMAG-UMRCNRS5149-c.c.051,UniversityofMontpellier, PL.E.Bataillon,F-34095Montpellier,France;boyom@math.univ-montp2.fr;Tel.: +33-467-143-571 † INMEMORIAMOFALEXANDERGROTHENDIECK.THEMAN. AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 1 June2016;Accepted: 16November2016;Published: 2December2016 Abstract:Letusbeginbyconsideringtwobooktitles:Aprovocativetitle,WhatIsaStatisticalModel? McCullagh(2002)andanalternativetitle, InaSearch forStructure. TheFisher Information.Gromov (2012). It is therichness inopenproblemsandthe linkswithotherresearchdomains thatmakearesearch topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statisticalmodels. This highlights the importance of both questions raised byPeterMcCullagh andMishaGromov. The title of thispaper looks like a list of keywords. However, the aim is to emphasize the linksbetween those topics. The theoryofhomologyofKoszul-Vinbergalgebroids and theirmodules (KVhomology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theoryofsemi-simplicialobjects. Thethirdconstruction isbasedontheanomalyfunctionsofabstract algebras and their abstractmodules. Weuse theKVhomology for investigating links between differential information geometry and differential topology. For instance, “dualistic relation of Amari”and“RiemannianorsymplecticFoliations”;“Koszulgeometry”and“linearizationofwebs”; “KVhomology”and“complexityofmodels”.Regardingthecomplexityofamodel, thechallenge is tomeasurehowfar frombeinganexponential family isagivenmodel. InPartAwedealwith theclassical theoryofmodels. PartB isdevotedtoansweringbothquestionsraisedbyMcCullagh andBGromov.Afewcriticismsandexamplesareusedtosupportourcriticismsandtomotivate anewapproach. Inagivencategoryanoutstandingchallenge is tofindaninvariantwhichencodes thepointsofamoduli space. InPartBweface fourchallenges. (1)The introductionofanewtheory ofstatisticalmodels. This re-establishmentmustanswerbothquestionsofMcCullaghandGromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphismclassofmodels; (3)The introductionof the theoryofhomological statisticalmodels. This is a pioneering notion. We address its linkswithHessian geometry; (4)We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theoryofhomological statisticalmodels. Subsequently, thedifferential informationgeometryhas ahomologicalnature. That isanothernotable featureofourapproach. Thispaper isdedicatedtoour friendandcolleagueAlexanderGrothendieck. Keywords:KVcohomoloy; functorofAmari;Riemannian foliation; symplectic foliation; entropy flow; moduli space of statistical models; homological statistical models; geometry of Koszul; localization;vanishingtheorem MSC:55R10;55N20;62B05;55C12;55C07 Entropy2016,18, 433 139 www.mdpi.com/journal/entropy