Page - 204 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Let (Ξ,Ω,p)beaprobabilityspace.ArandomHessianmetricQgeneratesaHessianstructure (M,gQ,D)whose tensormetricgQ isdefinedby gQ(X,Y)(x)= ∫ Ξ Q(x,ξ)(X,Y)p(ξ)dξ. ThegroupAut(Ξ,Ω)ofmeasurable isomorphismsof (Ξ,Ω) isdenotedbyΓ. Warning. WhenΞ is a topological space elements ofΓ are continuousmaps. WhenΞ is a differentiablemanifold elements ofΓaredifferentiablemaps. LetP(Ξ)be theBooleanalgebraof all subsets ofΞ. Theabstractgroup Aut(Ξ,P(Ξ)) is a subgroupof thegroup Isom(Ξ)of isomorphismsof the setΞ. Definition45. Ameasurable set (Ξ,Ω) is calledhomogeneous if thenatural actionofΓ inΞ is transitive. Throughout this paper we will be dealing with homogeneous measurable sets. Below we introduce the frameworkof the theoryofstatisticalmodels. 8.2. TheCategoryFB(Γ,Ξ) 8.2.1. TheObjectsofFB(Γ,Ξ) Definition46. Anobject of the categoryFB(Ξ,) is adatum [E,π,M,D]which is composedas it follows. (1) Misaconnectedm-dimensional smoothmanifold. Themap π :E→M isa locally trivialfiberbundlewhosefibersEx are isomorphic to the setΞ. (2) Thepair (M,D) is anm-dimensional locallyflatmanifold. (3) There is agroupaction Γ× [E×M]×Rm (γ,[e,x,θ])→ [[(γ ·e),γ ·x],γ˜ ·θ]∈ [E×M]×Rm. Thataction is subject to the compatibility requirement π(γ ·e)=γ ·π(e) ∀e∈E. (4) Everypoint x∈MhasanopenneighborhoodUwhich is thedomainof a localfiber chart ΦU×φU : [EU×U] (ex,x)→ [ΦU(ex),φU(x)]∈ [Rm×Ξ]×Rm. The local charts are subject to the followingcompatibility relation • (U,φU) is anaffine local chart of the locallyflatmanifold (M,D), • φU(π(e))= p1(ΦU(e)). (5) Weset ΦU(e)=(θU(e),ξU(e))∈Rm×Ξ. Let (U,Φ×φ)and (U∗,Φ∗×φ∗)be two local chartswith U∩U∗ =∅, 204