Seite - 163 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Nowadays, theredoesnot exist anycriterion fordecidingwhetheramanifoldsupports those differential topologicalobjects.Ouraimis todiscusssufficientconditions foramanifoldadmitting thosestructures.Ourapproach leads tonotable results. Thekeytoolsare theKVcohomologyandthe dualistic relationofAmari. BoththeKVcohomologyandthedualistic relationproduct remarkable split exact sequences. Notable results are basedon those exact sequences. HGEstands forHessian GEometry. Thepurposes:Hessianstructures,geometryofKoszul, hyperbolicity, cohomologicalvanishing theorems. Ouraims: ThegeometryofKoszul is a cohomologicalvanishing theorem. Statisticalgeometryandvanishing theorem, the solution toaholdquestionofAlexanderKGuts (announced). Theorem3asin[2]mayberephrasedintheframeworkofthetheoryofKVhomology.Foracompact locallyflatmanifold(M,∇)beinghyperbolic it isnecessaryandsufficient thatC2KV(A,C∞(M))containsa positivedefiniteEXACTcocycle. Tobehyperbolic isageometrical-topologicalpropertyof thedeveloping mapof locallyflatmanifolds. Tobehyperboliciticmeans that the imageof thedeveloping isaconvex domainnotcontaininganystraight line. This formulation is far frombeingahomological statement. So the Hessian GEOmetry is a link between the theory of KV homology and the Riemannian Riemanniangeometry. ThegeometryofKoszul, thegeometryofhomogeneousboundeddomainsandrelated topics havebeenstudiedbyVinberg,Piatecci-Shapiroandmanyothermathematicians [3]. Thegeometry ofSiegeldomainsbelongs to thatgalaxy[7,12].Almostallof thosestudiesarecloselyrelatedto the Hessiangeometry. Amongtheopenproblems in theHessiangeometryare twoquestionsweareconcernedwith. The first is to know whether the metric tensor g of a Riemannian manifold is a Hessian metric. AlexanderK.Guts raised this question in amail (tome) forty years ago. The secondquestion is to knowwhether a locally flatmanifold admits aHessian tensormetric. The solutions to those twoproblemsareannouncedin theAppendixAtothispaper. IGEstands for InformationGEometry. The purposes: The differential geometry of statistical models, the complexity of statistical models, ramificationsof the informationgeometry. Ouraims: Werevisit theclassical theoryofstatisticalmodels, requestsofMcCullaghandGromov.Asearch of a characteristic invariant. Themoduli spaceofmodels. Thehomologicalnatureof the informationgeometry. Theinformationgeometry is thedifferentialgeometry instatisticalmodels formeasurablesets. In both the theoretical statistics and the applied statistics the exponential families and their generalizations are optimal statisticalmodels. There aremany references, e.g., [17,18,22,37]. Here Murray-Rice1.15meansMurray-RiceChapter1,Section15.Amajorproblemis toknowwhethera givenstatisticalmodel is isomorphic toanexponentialmodel. That iswhatwecall thecomplexity problemof statisticalmodels. This challenge is a still openproblem. It explicitly arises from the purposeswhicharediscussed in [22]here, seealso [30]. In theappendix to thispaperwepresenta recentlydiscovered invariantwhichmeasureshowfar frombeinganexponential family isagiven model. That invariant isuseful forexploring thedifferential topologyof statisticalmodels. That is particularlyimportantwhenmodelsaresingular,vizmodelswhoseFisherinformationisnotinversible. ENTstands forENTropy. PierreBaudotandDanielBennequinrecentlydiscoveredthat theentropyfunctionhasahomological nature[31].Werecall that in2002PeterMcCullaghraisedafundamentalgeometric-topologicalquestion in the theoryof information:What IsaStatisticalModel? [30]AfewyearsafterMishaGromovraised asimilar request: TheSearchofStructure. Fisher Information[15,16]. Those twotitlesare twoformulationsof thesameneed. ThepaperofMcCullaghbecamethesubjectof controversy. Itgaverise toquestions,discussions, criticisms, see [30]. 163