Page - 214 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 beanm-dimensionalstatisticalmodelforameasurableset(Ξ,Ω). Todefinethefamilyofα-connections wework ina local chart (ΦU,ΦU). Weset ΘU=φU(U), Θ×Ξ=ΦU(EU). Inthebasemanifold(M,D) thelocalchart [U,φu)yieldsasystemoflocalaffinecoordinatefunctions θ=(θ1,...,θm). Weuse the notation as in [18]. Given a real number α wedefine the α-connection∇α by its Christoffel symbols in the local coordinate functions θ. ThoseChristoffel are denoted by Γαi j : k. Weproceedas it follows. Step1: In theopensubsetΘU⊂Rmweput Γ˜α,Uij:k (θ)= ∫ Ξ PU(θ,ξ) { [ ∂2ln(θ,ξ) ∂θi∂θj + 1+α 2 ∂ln(θ,ξ) ∂θi ∂ln(θ,ξ) ∂θj ] ∂ln(θ,ξ) ∂θk } dξ. This localdefinitionof Γ˜αij:k agreeswithaffinecoordinatechange inΘU. Step2: In theopensubsetUΓα,Uij:k isdefinedby Γα,Uij:k = Γ˜ α,U ij:k ◦φU. Since thedefinitionof Γ˜αij:k agreeswithanaffinecoordinatechangewecanuseanatlas A=[Uj,Φj×φj,γij] for constructing aKoszul connection∇α(A). Since the construction of∇α(A) agreeswith affine coordinatechangetheconnection∇α(A) is independent fromthechoiceofA. Everyα-connection is torsionfree. SoanMSE-fibration [E,p]→ [M,D] givesrise toama R α→∇α∈SLC(M). If theFisher informationg isdefinite then (M,g,∇α,∇−α) isadualpair [17,48]. By thevirtueof thedefinitionof theFisher informationga local sectionof sectionofKer(g) is a localvectorfieldX∈X(M)suchthat X ·p=0. Therefore, it iseasilyseenthat LXg=0. So ifdataareanalytic theng isastratifiedRiemannianfoliation. 8.4.4. TheHomologicalNatureof theProbabilityDensity WeconsideraMSE-fibration M :=[E,p]→ [M,D]. Werecall thatarandomdifferentialq-forminE isamapping E e→ω(e)∈∧q(T∗π(e))M 214