Page - 199 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Wehave introducedthedoublecomplex{ C : Cij=Ciτ(A∗,A∗)⊗Cjτ(A,C∞(M)), δij } . It gives rise to spectral sequenceswhichmaybeuseful for computing theKVcohomologies H∗τ (A∗,A∗) andH∗KV (A,C∞(M)). That is not thepurposeof thispaper. However thisdouble complex isuseful for replacingthefirstorderdifferentialequation D∇∇ ∗ (ψ)=0 bythehomologicalequation δ1,2qψ=0. WehaveprovedthehomologicalnatureofthespaceofgaugehomomorphismsM(∇,∇∗). This is useful for relatingthe imageofM(∇,∇∗) inH1τ(A∗,A∗)andthepairH2dR(M),H1,2(C). 8.B.TheTheoryofStatisticaLModels In the introductionof thispaperwehaverecalledtheproblemraisedbyPeterMcCullagh. What isastatisticalmodel [30]? Bythewaywehaverecalledavariant requestofMishaGromov. InaSearchforaStructure. TheFisher Information[15,16]. McCullaghandGromovchoose thesameframeworkforaddressingtheirpurpose,The theoryof category. ThisPartB isdevotedto thesamepurpose. Further themoduli space of isomorphism class of objects of a category C is denoted by [Ca]. Ingeneral it isdifficult tofindan invariant invawhichencodes [Ca]. Subsequently to thequestions raisedbyMcCullaghandbyGromov themoduli spaceof isomorphismclass of statisticalmodels is discussed in this Part B.Nowadays, there exists awell established theory of statisticalmodels. The classical references are Amari [17], Amari and Nagaoka [18]. Other remarkable references are Barndorff-Nielsen (Indian Journal ofMathematics 29, RamanujanCentenaryVolume) [21,24], KassandVos[37],Murray-Rice (Chapter1,Section15 in[22]). InPartAof thispaperwehavebeen dealingwith thiscurrent theory. Ithasbeencalled the local theory.Wesuggest readingtheattemptby McCullaghtoestablishaconceptuallyconsistent theoryofstatisticalmodels [30]. In its time, thepaper ofMcCullaghhadbeentheobjectofcontroversyandquestions. Weareaimedatre-establishingthe theoryofstatisticalmodels.Ourmotivationshaveemerged fromsomecriticisms. The current theory presents somedeficiencies thatweplan outlining. (i) Aweakness of the current theory is its lacking in geometry; (ii) In the literature on the information geometrymany referencesdefine anm-dimensional statisticalmodel as anopen subset of anEuclidean spaceRm. Though thisdefinitionmaybeuseful fordealingwithcoordinate functions, it is topologicallyand geometricallyuseless. LetΓbe thegroupofmeasurable isomorphismsof ameasurable set (Ξ,Ω). The informationgeometryof a statisticalmodelM includes thegeometry in the senseofErlangen programof thepair [M,Γ]. Let M and M∗ be m-dimensional statistical models for the same measurable set (Ξ,Ω). An isomorphismofM onM∗ looks like an sufficient statistic. The geometries [M,Γ] and [M∗,Γ] provide thesameinformation. So the impacton theapplied informationgeometryof the theoryof moduli space isnotable. Subsequently thesearch for characteristic invariantspresentsa challenge. An invariant is calledcharacteristic if itdetermineamodelup to isomorphism. Soacharacteristic invariant encodes themoduli space. That increases the interest in the search of bothMcCullagh andGromov. The Fisher information ofwidely usedmodels areHessianmetrics [52]. This observation is relevant.However theFisher information isnotacharacteristic invariant. 199