Page - 223 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 It is clear that Qj=Q◦Φ−1j . TheactionofΓ inE isQ-preserving. Thusahomological statisticalmodel isaquintuplet [E,π,M,D,Q]. HereQ isamap E e→Q(e)∈Z2KV(A,R). Thusweget randomcohomologicalmap E e→ [Q](e)= [Q(e)]∈H2KV(A,R). Definition63. Themapping [Q] is called cohomologymappingof thehomologicalmodel [E,π,M,D,Q]. 10.2.AnInterpretationof theEquivariantClass [Q] Weintendto interpret thecohomologyclass [Q]asanobstructionclass. Definition 64. (1) A homological statistical model whose cohomological map vanishes is called an EXact HomologicalStatisticalModel, (EXHSM); (2)Ahomological statisticalmodelwhosecocycle isarandomHessian metric iscalledaHEssianHomologicalStatisticalModel (HEHSM); (3)AnexactHessianhomological statistical model is calledaHYperbolicHomologicalStatisticalModel (HYHSM). GivenaHessianHomologicalmodel M=[E,π,M,D,Q] thecohomologymap [Q] is theobstructionforMbeinganHyperbolicitymodel. Thefollowingproposition leads to impactsonthedifferential topology. Proposition11. Thekernel of anexacthomological statisticalmodel is in involution. Further ifMandall data dependingonMareanalytic thenQisa stratified transversallyRiemannian foliation inM. If [E,π,M,D,Q] isexact thenthereexistsarandomdifferential1-formθ suchthat Q= δKVθ, viz Q(X,Y)=X ·θ(Y)−θ(DXY) ∀X,Y∈X(M). Thatuseful forseingthatKer(Q) is in involution. 10.3. LocalVanishingTheorems in theCategoryHSM(Ξ,Ω) Reminder. Thecategorywhoseobjects arehomological statisticalmodels (for (Ξ,Ω)) isdenotedbyHSM(Ξ,Ω). Henceforthwefixanauxiliary structureofprobability space (Ξ,Ω,p∗). Definition65. Weare interested inrandomfunctionsdefined inRm×Ξ. 223