Page - 225 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Bythevirtueof theLemmaofPoincaré thereexistsa local functionhj(θ,ξ)suchthat βj=dhj. Nowthedifferential1-form θ˜ isdefinedby θ˜=∑ j hj(θ,ξ)dθj. Direct calculations leadto the followingequality Q= δKVθ˜. Thisends theproofof (1). Theproofof (2). WeassumethatM=[E,π,M,D,Q]hasthepropertyp∗−Exp.Wekeepthenotation we justused. The randomdifferential 1-form θ˜ is a (deRham)cocycle. ThereforeΘ×Ξ supports a random functionh(θ,ξ)suchthat θ˜=dh. Sowehavethe followingconclusion Q(θ(e),ξ(e))=D2h(θ(e),ξ(e)) ∀e∈EU. Equivalentlyonegets ∂2h ∂θi∂θj =Qij. Since M has the property p∗−Exp we choose a function h has the property p∗−EXP. The functionsF(θ)andP(θ,ξ)aredefinedby F(θ)= ∫ Ξ exp(h(θ,ξ))dp∗(ξ), PQ(θ,ξ)= exp(h(θ,ξ)) F(θ) . By thevirtueof theproperty p∗−Exp the functionP(θ,ξ)satisfies the followingrequirements (i) PQθ,ξ) isdifferentiablewithrespect toθ, (ii) PQ satisfies the following inequalities 0≤PQ(θ,ξ)≤1, (iii)PQ satifies the following identity ∫ Ξ PQ(θ,ξ)dξ=1. Thusthepair(ΘU,PQ) isalocalstatisticalmodelfor(Ξ,Ω). Thisendstheproofof(2). Thetheorem isdemonstrated. Thepair (ΘU,PQ) is calleda localizationofM. Definition66. Alocalization (ΘU,PQ) is calledaLocalVanishingTheoremof [E,π,M,Q]. 225