Page - 216 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 This isanother topologicalnatureof theentropy. Foranotherviewpointsee [16,31]. Our purpose is to show the theory of statistical models has a homological nature in the categoryFB(Γ,Ξ). Definition59. Astatisticalmodel for ameasurable set (Ξ,Ω) is couple (M,[p]) formedby anobject of the category (FB−Γ,Ξ), namely M=[E,π,M,D] andasmoothΓ-equivariant randomcohomologyclass [p]∈H0e(E,R). Further the to everyfiber p|Ex is aprobabilitydensity. AComment. Let (U,Φ×φ)bea local chart of [E,π,M,D]and let x∗∈U.Weset ΘU=φ(U), (Ex∗,Ωx∗)=Φ−1[{φ(x∗)}×(Ξ,Ω)]. The lastdefinitionabove says thatweobtain theprobability space (Ex∗,Ωx∗,[p]). Thispropertydoesnotdependonthe choiceof the local chart (U,Φ×φ). Thuswecanregard [M,p]asa special typeofhomologicalmap FB(Γ,Ξ) M→ [p]∈H0e(E,R). 9.TheModuliSpaceof theStatisticalModels Wearegoing to faceanothermajoropenproblem. Thechallenge is thesearch foran invariant whichencodes thepointsof theorbit space m= M G . That iswhatiscalledtheproblemofmodulispace.Thisproblemofmodulispaceisamajorchallenge inboththedifferentialgeometryandthealgebraicgeometry(seethetheoryofTeichmuller).Theproblem isratherconfusedlyaddressed in [30]. Subsequently itprovokedcontroversiesandcriticisms. TheHessianFunctor Weconsider thecategoryBFwhoseobjectsarepairs{M,B} formedbyamanifoldMequipped bilinear formsB∈Γ(T∗⊗2M). InPartAwehavedefinedtheHessiandifferentialoperatorofaKoszulconnection∇, namely D∇=∇2. Thoseoperatorsareuseful foraddressingtheproblemofmoduli spaces. Forourpurpose four categoriesare involved, (1) ThecategoryLCwhoseobjectsaregaugestructures (M,∇), (2) ThecategoryGMwhoseobjectsarestatisticalmodels formeasurablesets, (3) thecategoryBFwhoseobjectsaremanifoldsequippedbilinear forms, 216