Page - 224 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (1) Arandomfunction f has theproperty p∗−EXPif exp(f(x,ξ))≤ ∫ Ξ exp(f(x,ξ))dp∗(ξ) ∀x∈Rm. (2) Arandomcloseddifferential1-formθhasthepropertyp∗−EXPifeveryx∈Rmhasanopenneighbourhood Usatisfying the followingconditions,U×Ξ support a randomfunction f subject to tworequirements: • θ=df, • f has theproperty p∗−Exp. (3) An exact homological statisticalmodel [E,π,M,D,Q] has property p∗−EXP if there exists a random differential1-formθ satisfying the followingconditions • θhas theproperty p∗−EXP, • Q= δKVθ. Localization Ourpurpose is toexplore therelationshipsbetweenthe theoryofhomological statisticalmodels andthe theoryof local statisticalmodelas in [18,22],Barndorff-Nielsen1987 Ouraimis toshowthat thecurrent (local) theory isabyproductof the localizationofhomological models. Thenotionof localizationofhomologicalmodels isbut thenotionof localvanishingtheorem. Theorem22. Let [E,π,M,D,Q]beahomological statisticalmodel. (1) [E,π,M,D,Q] is locally exact. (2) If the [E,π,M,D,Q]has theproperty p∗−EXPthen [E,π,M,D,Q] is locally isomorphic toaclassical statisticalmodel (Θ,P)as in [18]. TheSketchofProofof (1). Let (U,Φ×φ)bea local chartof [E,π,M,Q].Weset ΘU=φ(U). We assume that ΘU is an open convex subset of Rm. Θ supports a system of affine coordinate functions θ=(θ1,...,θm). Wehave Q(θ,ξ)=∑Qij(θ,ξ)dθidθj. SinceQ(θ,ξ) isa randomKVcocycleof A˜wehave δKVQ=0. The lastequality isequivalent to the followingsystem ∂Qjk ∂θi − ∂Qik ∂θj =0. Wefixξ∈Ξ. Forevery j therandomdifferential1-formβj isdefinedby βj(θ,ξ)=∑ i Qijdθi. Everyβj(θ,ξ) isacocycleof thedeRhamcomplexofΘU. 224