Page - 144 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 adoptedthreeapproaches. Eachapproachisbasedonitsspecificmachinery.However, thereaderswill face threecochaincomplexeswhicharepairwisequasi isomorphic. TheKVcohomologyispresent throughoutthispaper.AttheendofPartBthereaderwillseethat thetheoryofstatisticalmodels isbut avanishingtheoreminthe theoryofKVhohomology. Thefirstapproach isbasedonthepioneering fundamentalbrute formulaof thecoboundaryoperator.Historically, thebrute formula is thefirst to havebeenconstructed[9]. Thisfirstapproach isusedinmanysectionsof thispaper. Regardingthe theoryofdeformation of theKoszulGeometry, theKVcohomologyis thesolutionto theconjectureofGerstenhaber. In the theoryofmodulesofKValgebroids theroleplayedbytheKVcohomology ispracticallySIMILAR to theroleplayedbytheHocshildcohomology in thecategoryofassociativealggebroidsandtheir modules. This last remarkholds for theroleplayedbytheChevalley-Eilenbergcohomology in the categoryofLie algebroids and theirmodules. Nevertheless, our comparison fails in the theoryof Extensionofmodulesoveralgebroids. Inbothcategoriesofextensionsofmodulesoverassociative algebroidsandLiealgebroids themoduli spaceofequivalenceclass isencodedbycohomologyclasses ofdegreeone. In thecategoryofextensionsofKVmodules themoduli space isencodedbyaspectral sequence. Thatwasaunexpectedfeature in [9]. ThepioneeringcoboundaryoperatorofNijenhuis [28] maybederivedfromthetotalbrutecoboundaryoperator introducedin[29]. Thesecondapproach isbasedonthenotionofsimplecialobjects. Thethirdapproachisbasedonthetheoryofanomalyfunctions forabstractalgebrasandtheir abstractmodules. The ideahasemergedfromrecentcorrespondenceswithoneofmyformer teachers. TheKVanomaly functionof aKoszul connection∇maybe expressed in termsof the∇-Hessian operators∇2,namely KV∇(X,Y,Z)=<∇2(Z),(X,Y)>−<∇2(Z),(Y,X)> . This approach is a powerful for addressing the relationships between the global analysis, the differential topology and the information geometry. The approach by the anomaly functions suggestsmanyconjectures.Amongthoseconjectures is the following. Conjecture. Everyanomaly functionof algebrasandofmodulesyieldsa theoryof cohomologyof algebras andmodules. Section3. This section isdevotedto the theoryofKV(co)homologyofKoszul-Vinbergalgebroids. Wefocusoncohomologicaldatawhichareusedin thepaper. Section4. This section isdevoted theKValgebroidswhicharedefinedbystructuresof locally flatmanifold. TheKVcohomology theory is used forhighlighting the impacts on thedifferential topologyof the informationgeometry and itsmethods. Wemake themost of some relationships betweentheKVcohomologyandtheglobalanalysisof thedifferentialequationFE∗(∇∇∗).Wealso sketch theglobalanalysisof thedifferentialequation FE∗∗(∇). This leads to the function LC ∇→ rb(∇)∈Z. Weexplainhowto interpret rb asadistance. (See theAppendixAtothispaper ). For instance, the function rb gives rise to an numerical invariant rb(M)which measures how far from being anexponential familyisastatisticalmodelM. Thisresult isasignificantcontributiontotheinformation geometry, see [18,22,24]. Section5.Weare interested inhowinteract the informationgeometry, theKVcohomologyand thegeometryandKoszul. Inparticularwerelate thenotionofhyperbolicityandvanishingtheorems in theKVcohomology. 144