Page - 226 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Let [E,π,M,D,p]beanobjectof thecategoryGM(Ξ,Ω).Weset Qp=D2log(p). Theforeweget theexacthomological statisticalmodel Mp=[E,π,M,D,Qp] So the notion of vanishing theorem has significant impacts on the information geometry. To simplify an exactmodelswhich having the property p∗−Exp (for someprobabily space) are iscalledExp−models. Theorem23. Thenotation is thatusedpreviously. (1) ThecategoryGM(Ξ,Ω) is a subcategoryof the categoryEXHSM(Ξ,Ω). (2) Ojects ofGM(Ξ,Ω)buthomologicalExp−models. Reminder:NewInsigts. (1.1) The InformationGEometry is thegeometryof statisticalmodels. (1.2) The Information topology is the topologyof statisticalmodels. (2.1) Thehomologicalnatureof the InformationGeometry. (2.2) What is a statisticalmodel? Theanswer to thequestionraisedbyMcCullaghshouldbe:Astatisticalmodel is aGlobalVanishingTheoremin the theoryofhomologicalmodels. (2.3) A local statisticalmodel is aLocalVanishingTheoremin the theoryofhomologicalmodels. 11.TheHomologicalStatisticalModelsandtheGeometryofKoszul Our purpose is to to relate the category of homological statisticalmodels and the geometry ofKoszul. This relationship isbasedonthe localizationofhomological statisticalmodels. Proposition12. EXPHEHSM(Ξ,Ω) stands for the subcategorywhoseobjects areHessianExp-models. (1) Theholomogicalmap leads to the functorofEXPHEHSM(ß,Ω) in the categoryofHessianstructures in (M,D) [E,π,M,D,Q]→ (M,D,Q˜). (2) IfMis compact then the subcategoryof exactHessianhomologicalExp-modelsEXPHYHSM(Ξ,Ω) is sent in the categoryofhyperbolic structure in in (M,D). 12. Examples Thissection isdevotedtoafewexamples. Theconstruction involvessomebasicnotionsof the differential topology. Example1:Dynamics Weconsidera triple [M×H,p1,M,∇].Here (M,∇) isacompact locallyflatmanifold, (H,dμ) is anamenablegroup. There isaneffectiveaffineaction H×(M,∇)→ (M,∇). Let f ∈C∞(M)andx∈M. The function fx :H→R 226