Page - 212 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 subject to therequirement D2 ∫ F a=0. Definition57. AnMSE-fibration [E,p]→ [M,D] is calledanexponentialmodel if the followingconditionsare satisfied. (1) ThebasemanifoldMsupports a locallyflat structure (M,∇)andarealvalued functionψ∈C∞(M). (2) The total spaceE supports a realvaluedrandomfunction a. (3) The triple [a,∇,ψ] is subject to the followingrequirement (4) ∇2∫F(a)=O, (5) p(e)= exp[a(e)−ψ(π(e))]. Remark5. AtonesideLocalizationsof exactHessianhomological statisticalmodelsyield the classicalKoszul InformationGeometry [55]. That is but the classicalHessian InformationGeometry. Atanother side theKV homologylearnsthat theHessian informationGeometry is thesamethinkas thegeometryof exponential famillies seePartASection5,Theorem16. Reminder. In theAppendixAto thispaper the readerwillfindanewinvariant rb(p)measuringhowfar frombeing anexponentialmodel is anMSE-fibration [E,p]→ [M,D]. Bythevirtueof results inPartA, tobeanexponentialmodeldependsonhomological conditions. 8.4.1. TheEntropyFlow Wearegoingto introduce thenotionof localentropyflow. Subsequentlywewill showthat the Fisher informationofamodel [E,π,M,D,p] is theHessianof the localentropyflow. TostartweconsideraMSE-fibration [E,p]→ [M,D]. That isanotherpresentationof thestatisticalmodel [E,π,M,D,p]. Let [Uj,Φj×φj,γij,Pj]beanatlasof [E,π,M,D,p].Weput [Θj,Pj]= [Φj(Ej),p◦Φ−1j ]. Thenevery [Θj,Pj] isa local statisticalmodel for (Ξ,Ω). LeX,Ybe twovectorfieldsdefined inUj and letψX(t)andψY(s)be their localflowsdefined inUj. Thenweset Φj(e)= [θj(e),ξj(e)]= [φj(π(e)),ξj(e)], e∈Ej, ψ˜j(t)[θj(e),ξj(e)]= ([φjψX(t)φ−1j ][θj(e)],ξj(e)), ψ˜j(s)[θj(e),ξj(e)]= ([φjψY(s)φ−1j][θj(e)],ξj(e)). Definition58. The local entropyflowof thepair (X,Y) is the functionEntjX,Y definedby EntjX,Y(s,t)(π(e))= ∫ Ξ { Pj[ψ˜X(s)(Φ(e))]log[Pj[ψ˜Y(t)(Φj(e))]] } dξ(e). 212