Page - 196 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 The identities above showthat ∂2a ∂θi∂θj (θ0)=0 for allθ0∈U.Thereby the function θ→ψ(θ)+ log(P(θ,ξ)) dependsaffinelyonθ1,...,θm. So there exists aRm+1-valued function Ξ ξ→ (C(ξ),F1(ξ),...,Fm(ξ))∈Rm+1 such that ψ(θ)+ log(P(θ,ξ))=C(ξ)+ m ∑ 1 Fi(ξ)θi. Infinalweget P(θ,ξ)= exp(C(ξ)+∑Fi(ξ)θi−ψ(θ)) for all (θ,ξ)∈U×Ξ. So (Θ,P) is locallyanexponential family. Since the exponential function is injective this local property of (Θ,P) is a global property, in other words themodel is globally an exponentialmodel. In final assertion (1) implies assertion(2). This ends the demonstrationof the theorem SomeComments. (i) Itmust be noticed that the demonstration above is independent of the rank of the Fisher information g. Therefore, the theoremholds insingular statisticalmodels. (ii) In regular statisticalmodels the theoremabove leads to thenotionof e-m-flatnessas in [18]. (iii)When the Fisher information g is semi-definite the dualistic relation is meaningless. However data (Θ,g,∇,∇∗)mayberegardedasdatadependingonthe transversal structureof thedistributionKer(g). (iv) In theanalytic category theFisher information isaRiemannian foliation. Therefore, both the information geometry and the topology of information are transversal concepts. Thismay be called the transversal geometryandthe transversal topologyofFisher-Riemannian foliations. (v) The theoremabovedoesnot solve thequestionashowfar frombeinganexponential family is agivenmodel. It only tellsus that exponential families areobjects of theHessiangeometry. Theframeworkforaddressingthechallenge justmentionedis the theoryof invariants. That is the purposeofa forthgoingwork. Somenewresultsareanouncedin theAppendixAtothispaper. 6.TheSimilarityStructureandtheHyperbolicity Weconsideraduallyflatpair(M,g,∇,∇∗). Both(M,∇,g)and(M,g,∇∗)are locallyhyperbolic in thesenseof [2]. So they locallysupport thegeometryofKoszul. That isaconsequenceof theclassical LemmaofPoincare. EverypointofMhasanopenneighborhoodUsupportingalocaldeRhamcloseddifferential1-forms ω∈C1KV(A,R) and ω∗ ∈C1KV(A∗,R) subject to the followingrequirements g|U= δω, g|U= δ∗ω∗. 196