Page - 206 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 8.2.2. TheMorphismsofFB(Γ,Ξ) Let [E,π,M,D]and [E∗,π∗,M∗,D∗]betwoobjectsofFB(Γ,Ξ). LetΨ×ψbeamap [E×M] (e,x)→ (Ψ(e),ψ(x))∈ [E∗×M∗]. Definition47. Apair (Ψ×ψ) is amorphismof thecategoryFB(Γ,Ξ) if the followingconditionsare satisfied (m.1): π∗◦Ψ=ψ◦π, (m.2): bothΨandψareΓ-equivariant isomorphism, that is to say Ψ(γ ·e)=γ ·Ψ(e), ψ(γ ·x)=γ ·ψ(x), (m.3): ψ is anaffinemapof (M,D) in (M∗,D∗). TheFigure3 represents theproperties (m.1)and(m.2). Wearegoing todefine thecategoryof statisticalmodel for (Ξ,Ω). The framework is thecategoryFB(Γ,Ξ). B1A1 C1 A B C p1π p1 φu Φu∗ Φu γuu∗ γuu∗ φu∗ Figure3.Equivariance. Atonesidewerecall that thegroupΓalsoacts inRm×Ξ.Atanotherside the localizationsare madecoherent thanks toCechcocyclesγUU∗. Figure3 tells twoinformations. Firstly localizationsare Γ-equivariant, secondly thanks toCechcocycles localizationsarecoherent. 8.3. TheCategoryGM(Ξ,Ω) Wekeepthenotationusedin theprevioussubsections.Ourpurpose is thecategoryofstatistical modelsGM(Ξ,Ω). 8.3.1. TheObjectsofGM(Ξ,Ω) Definition48. Anm-dimensional statistacalmodel for (Ξ,Ω) is anobject ofFB(Γ,Ξ), namely M=[E,π,M,D] whichhas the followingproperties (ρ∗). [ρ1]: For every local chart (U,ΦU×φU) the subset [ΘU×Ξ]=ΦU(EU) 206