Page - 189 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Theconsiderationswejustdiscussedmayhaveremarkableimpactsonthetopological-geometrical structureofstatisticalmanifolds. Fromourbriefdiscussionweconclude Theorem12. Consider a statisticalmanifold (M,g,∇,∇∗). Every non trivial simple symplectic foliation ω∈Ω∇2 (M) isdefinedbyaRiemanniansubmersiononasymplecticmanifold. Corollary4. Everynontrivial simple symplecticweb [ωj, j∈ J]⊂Ω∇2 (M) isdefinedby familyofRiemannian submersionsonsymplecticmanifols. 5.2. TheHessian InformationGeometry,Continued In [52]Shimapointedout that theFisher informationsofmanyclassical statisticalmodels are Hessianmetric tensors. Atanother side theexponentialmodels (orexponential family)maybeconsideredasoptimal Statisticalmodels. Asalreadymentionedtheredoesnotexistsanycriterionforknowingwhetheragivenstatistical model is isomorphic toanexponentialmodel [22], [13] In thecategoryof regularmodels,vizmodelswhoseFisher information isaRiemannianmetric, it isknownthat theFisher informationofanexponentialmodel isaHessianRiemannianmetric [18,52]. In this subsectionweaddress thegeneral situation.Wegiveacohomological characterizationof exponentialmodels.Wealso introduceanewnumerical invariant rbwhichmeasureshowfar from beinganexponential family isagivenstatisticalmodel. See theAppendixAtothispaper. Werecall that themetric tensorgofaHessianstructure (M,D,g) isa2-cocycleof theKVcomplex [C∗KV(A,R),δKV]. Tononspecialistswegotorecallsomeconstructions inthegeometryofKoszul [2,52], seealso[53]. Let ((M,x∗),D)beapointed locallyflatmanifoldwhoseuniversal covering isdenotedby (M˜,D˜). Here the topological space M˜ is thesetofhomotopyclassofcontinuouspaths {([0,1],0)→ (M,x∗)}. Its topology is thecompact-opentopology. Let cbeacontinuouspathwith c(0)=x∗. For s∈ [0,1] theparallel transportofTx∗M inTc(s)M isdenotedbyτs. OnedefinesQ(c)∈Tx∗Mby Q(c)= ∫ 1 0 τ−1s ( dc(s) ds )ds. The tangentvectorQ(c)dependsonlyonthehomotopyclassof thepath c. Therefore,Qdefines a localhomeomorphism Q˜ : M˜→Tx∗M. Letπ1(x∗)bethe fundamentalgroupatx∗. Let [γ]∈π1(x∗). Thenatural leftaction π1(x∗)×M˜→ M˜ isgivenbythecompositionofpaths,viz [γ].[c]= [γ◦c]. Theparallel transportalonga loopγ(t)yieldsa linearactionofπ1(x∗) inTx∗Mwhich isdenoted by f([γ]). 189