Page - 213 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Topursueweuse thecompatibilityof local chartsof theatlas[Uj,Φj×φ,,γij,Pj] . If Ui∩Uj =∅ thenforall e∈EUi∩Uj wehave Φj(e)=γij ·Φi(e), φj(π(e))=γij ·φi(π(e)). Werecall that ψ˜(t)and ψ˜(s)aredefinedby ψ˜j(t)=φjψX(t)φ−1j , ψ˜j(s)=φjψY(s)φ−1j . Thoseremindersareuseful forconcludingthatwhenever Ui∩Uj =∅ wehave EntiX,Y(s,t)(π(e))=Ent j X,Y(s,t)(π(e)). So the localentropyflowdoesnotdependonlocal charts. IfbothXandYarecompletevectorfields thentheirentropyflowisgloballydefined.Anotable consequence is the followingstatement. Theorem18. EveryMSE-fibrationoveracompactmanifoldMadmitsagloballydefinedentropymap X(M)×X(M) (X,Y)→EntX,Y∈C∞(R2). 8.4.2. TheFisher Informationas theHessianof theLocalEntropyFlow weconsider the function Hj(s,t,ξ)=Pj[ψ˜X(s)(φ(e))]log[Pj[ψ˜j(t)(Φj(e))]]. Direct calculationsyield [ ∂2(Hj(s,t)) ∂s∂t ][(s,t)=(0,0)]=Pj[φj(e)](X · log[Pj(Φj(e))])(Y · log[Pj(φj(e))]). Weknowthat ∂ 2 ∂s∂t commuteswith ∫ Ξ. Therbyweconcludethat gπ(e)(X,Y)= ∂2Entj(s,t)(π(e)) ∂s∂t [(s,t)=(0,0)]. Theorem19. WeconsideranMSE-firationoveracompactmanifold M :=[E,p]→ [M,D]. TheFisher informationofM is theHessianof the entropymap. 8.4.3. TheAmari-ChentsovConnections inGM(Ξ,Ω) Let M=[E,π,M,D,p] 213