Page - 191 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 5.3. Theα-ConnetionsofChentsov Still, nowadays, the information geometry deals with models (Θ,P) whose underlying m-dimensionalmanifoldΘ isanopensubsetof theeuclideanspaceRm.Further theFisher information g isassumedtoberegular,viz (Θ,g) isaRiemannianmanifold. In thispaper thisclassical information geometry iscalledthe local informationgeometry. This“localnature”willbeexplainedinPartBof thispaper. At themomentweplanto investigateother topologicalpropertiesof the local statisticalmodels. Let(Θ,P)beanm-dimensional local statisticalmodel forameasurableset(Ξ,Ω). ThemanifoldΘ isadomain in theEuclideanspaceRm.The functionP isnonnegative. It isdefinedinΘ×Ξ.Werecall therequirementsP is subject to. (1) ∀ξ∈Ξ the function Θ θ→P(θ,ξ) is smooth. (2) ∀θ∈Θ the triple (Ξ,Ω,P(θ,−)) isaprobabilityspace. (3) ∀θ,θ∗∈Θwithθ = θ∗ thereexistsξ∈Ξsuchthat P(θ,ξ) =P(θ′,ξ). (with therequirement (3) (Θ,P) is called identifiable.) (4) Thedifferentiationdθ commuteswiththeintegration ∫ Ξ .TheFisher informationofamodel(Θ,P) is thesymmetricbi-linear formgwhich isdefinedby g(X,Y)(θ)= ∫ Ξ P(θ,ξ)[[dθlog(P)]⊗2(X,Y)](θ,ξ)dξ. Heredθ stands for thedifferentiationwithrespect to theargumentθ∈Θ. (5) TheFisher information ispositivedefinite. Remark 3. The Fisher information g can be defined using any Koszul connection∇ according to the following formula g(X,Y)(θ)=− ∫ Ξ P(θ,ξ)[(∇2log(P))(X,Y)](θ,ξ)]dξ. Therightmemberof the last equalitydoesnotdependonthe choice of theKoszul connection∇. Fromnowon,we dealwith a generic statisticalmodel. Thismeans thatwe do not assume the Fisher informationg isdefinite. Letθ=(θ1,...,θm)beasystemofEuclideancoordinate functions inRm.Toeveryrealnumberα is assigned the torsion freeKoszul connection∇α whoseChristoffel symbols in the coordinate (θj)are Γαij,k= ∫ Ξ P(θ,ξ)[( ∂2log(P(θ,ξ)) ∂θi∂θj + 1−α 2 ∂log(P(θ,ξ)) ∂θi ∂log(P(θ,ξ)) ∂θj ) ∂log(p(θ,ξ)) ∂θk ]dξ. Thisdefinitionagreeswithanyaffinecoordinate change.Weput∂i= ∂∂θi .Wehave ∇α∂i∂j=∑ k Γαij,k∂k. 191