Page - 192 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Nowweassumeamodel (Θ,P) is regular. Then theChristoffel symbols and theFisher informationare relatedby the formula Γαij,k=g(∇α∂i∂j,∂k). Further everyquadruple (Θ,g,∇α,∇−α) is a statisticalmanifold [18,48]. Thuswehavea familyof splittingshort exact sequences 0→Ω∇α(Θ)→M(∇−α,∇α)→S∇α2 (Θ)→0. Sothemachinerywehavedeveloped in theprecedingsectionscanbeperformed to explore thedifferential topologyof regular local statisticalmodels. For thatpurpose the crucial tool is the familyofvector space Sα2(Θ)=S∇ α 2 (Θ). Weconsider theabstract trivial bundle ∪α[Sα×{α}]→R whosefiberoverα∈R isSα(Θ). To everyB∈Sα(Θ)weassign theuniqueψ+∈Σ(g)definedby g(ψ+α(X),Y)=B(X,Y). Themachinery in theprecedingsubsection leads to the followingproposition. Proposition8. Weassume (Θ,P) is regular. (1) Everynonzero singular section R α→Bα∈Sα(Θ) gives rise the familyof (g-orthogonal)2-web TΘ=Ker(ψ+α)⊕ im(ψ+α). Furtheraccording to thenotationusedpreviously (Bα) is a familyofRiemannian foliationsas in [39,40]. (2) ByreplacingSα(Θ)byΩ∇2 (Θ) everynonzero singular section R α→ωα∈Ω∇2 (Θ) yieldsa familyof symplectic foliationsωα. Reminder. (i) α→Bα is calleda singular section if eachBα isnon inversible. (ii) α→ωα is calleda simple section if eachωα is simple. Wehaveusedsomegaugemorphisms toconstructRiemanniansubmersionsof statisticalmanifoldsover symplecticmanifolds. Thenotionswe just introduced lead to similar situations. Theorem15. Let (Θ,P)bea regular statisticalmodelwhoseFisher information isdenotedbyg. Everysimple nonzero singular section R α→ωα∈Ωα(Θ) definesanα-familyofRiemanniansubmersionsof (Θ,g)onto symplecticmanifolds. 192