Page - 229 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Step1 In thepair (F1,gf) theopensubsetU1⊂F1 isdefinedby x∈U1 if f rank(gf(x))= max [x∗∈F1] rank(gf(x∗)). Step2 We iterate this construction. Thenwehaveget afiltrationofM ...⊂Fn⊂Fn−1⊂ ...⊂F1⊂FO=M. Thisfiltrationhas the followingproperties (1) Fj−1\Fj is aanalytic submanifoldofM. (2) gf definesa regularRiemannian foliation in (Fj−1\Fj,gf), Remark6. Theextrinsicgeometryof submanifolds isaparticularcaseof thegeometryof singular foliation[25]. 13.HighlightingConclusions 13.1. Criticisms InPartBwehaveraisedsomecriticisms.Wehaveconstructedstructuresof statisticalmodels inflat tori. Anm-dimensionalflat torus isnothomeomorphic toanopensubsetofRm. Thesecond criticism is the lack of dynamics. Subsequently, the problemofmoduli space is absent from the classical theory. Thatdeficiency isfilled inbythecharacteristic functor M=[E,π,M,D,p]→ qM. Thecurrent theoryrequiresamodel tobe identifiable. Fromtheviewpointof locally trivialfiber bundles, that requirement isuseless. 13.2. Complexity In both the theoretical information geometry and the applied information the exponential models and their generalizations play notable roles. What we call the complexity of a model [E,π,M,D,p] is its distance from the category of exponential models. Up to today there does not exist any INVARIANTwhichmeasures how far frombeing an exponential is a givenmodel. This problemhas a homological nature. Wehave produced a function rb which fills in that lack. (See theAppendixAbelow). 13.3. KVHomologyandLocalization Wehave introducedthe theoryofhomologicalmodel.Amongthenotablenotions thatwehave studied is the localizationofhomological statisticalmodels. It links the theoryofhomologicalmodels andthecurrent theoryas in [22]. Itmaybe interpretedasa functor fromthe theoryofhomological models to theclassical theoryofstatisticalmodels. 13.4. TheHomologicalNatureof the InformationGeometry GM(Ξ,Ω) andHSM(Ξ,Ω) are introduced in this Part B. The category of local statistical models for (Ξ,Ω) isdenotedbyLM(Ξ,Ω).Ononeside, therightarrowsbelowmeansubcategory. Thenwehave LM(Ξ,Ω)→GM(Ξ,Ω)→HSM(Ξ,Ω). 229