Page - 193 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Proposition9. Weassumeamodel (Θ,P) is regular. For everynonzero realnumberαonehas M(∇α,∇−α)∩M(∇−α,∇α)=M(g,∇−α,∇α)+M(g,∇∗,∇). TheSketchofProof. Theproof isbasedontheshortexact sequences 0→M(g,∇α,∇−α)→M(∇α,∇−α)→S∇2 (Θ)→0, 0→M(g,∇−α,∇α)→M(∇−α,∇α)→S−∇2 (Θ)→0. Letussuppose that theconclusionof the theproposition fails. Thenthere isanonzero2-form B∈S∇α2 (Θ)∩S∇ −α 2 (Θ). (1) IfKer(B)=0, thenboth∇α and∇−α coincidewith theLevi-CivitaconnectionofB. This implies α=0, thiscontradictsourchoiceofα. (2) If Ker(B) = 0 then Ker(B) and Ker(B)⊥ are geodesic for both∇α and∇−α. Thus the pair (Ker(B),Ker(B)⊥)definesag-orthogonal2-web. Atoneside,by thevirtueof the reduction theoremas in [18]every leafFofKer(B)⊥ inheritsa dualpair (F,gF,∇αF,∇−αF ). Atanotherside,Bgivesrise to theRiemannianstructure (F,B). Furthermoreboth∇αF and∇−αF are torsionfreemetricconnections in (F,B). Therebyonegets ∇αF=∇−αF The last equalityholds if andonly ifα= 0.This contradictsourassumption. Theproposition isproved. Thepropositionabove isaseparationcriterionforα-connections in the followingsense. Forevery nonzerorealnumberα thevectorsubspaceS∇α2 (Θ) is transverse toS∇ −α 2 (Θ) in thevectorspaceS2(Θ) ofsymmetric forms inΘ. 5.4. TheExponentialModels and theHyperbolicity Achallengeisthesearchofacriterionfordecidingwhetheramodel(Θ,P) isanexponentialfamily. That is thechallenge in [22]. Still,nowadays, thisproblemisopen. TheFisher informationofaregularexponentialmodel isaHessianRiemannianmetric.Weare goingtodemonstratethat theconverseisgloballytrue.Ourproof isbasedoncohomologicalarguments. In theAppendixAtothispaperweintroduceanewnumerical invariantrb(Θ,P)whichmeasures howfar frombeinganexponential family isamodel (Θ,P). The invariant rbderives fromtheglobalanalysisofdifferentialoperators Dα=D∇α, Dα=D∇ α . Nowwearegoingtoprovideacohomological characterizationofexponentialmodels. Beforepursuingwerecalladefinition. Letθj, j :=1,...,mbeasystemofEuclideancoordinate functionsofRm. Definition40. [18]Anm-dimensional statisticalmodel (Θ,P) is called an exponentialmodel for (Ξ,Ω) if there exist amap Ξ ξ→ [C(ξ),F1(ξ),...,Fm(ξ)]∈Rm+1 193