Page - 146 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 This theorememphasizes theSearchforstructure [16]. Section10. This section isdevoted to introduce thecategoryofhomological statisticalmodels. Thismaybe interpretedasavariantof the topologyof the information. Anotherapproach is tobe foundinBaudot-Bennequin[31]. Thecurrent theory (as in [17]) is called theclassical (or local) theory. Thismeans thatastatistical modelas in [17,18] isderivedfromthe localizationofahomologicalmodel. Looselyspeakingsucha modelexpressesa localvanishingtheoreminthe theoryofhomological statisticalmodels. Section11. Thissection isdevotedtodiscussingthe linksbetweenthegeometryofKoszuland the theoryofhomological statisticalmodels. Those investigations leadto thisnotable feature. TheGeometryofKoszul, thehomologicalstatisticalmodelsandtheclassical informationgeometry locally lookalike. Section12. ThroughSection9theframeworkis thecategoryofequivariant locally trivialfibration. Thisassumption isweakenedinSection12.Werecall therelationshipsbetweentheCechcohomology andthe theoryof locally trivialfiberbundle.Weextendthescopeofapplicationsof themethodsof the informationgeometry. Thoseextensionsproducesomeinterestingresults.Here isan instance. Theorem2. LetMbeanorientedcompact real analyticmanifoldand letCω(M2)be the spaceof realvalued analytic functions in M2. There exists a non trivial map of Cω(M2) in the family of (positive) stratified Riemannian foliation inM. Sections13. ThisSection13 isavariantofSection7. Section 14 is an appendix we have mentioned. It is devoted to overview a few new significant results. Thoseresultsarederivedfromtheglobalanalysisof thedifferentialoperators{ D∇,D∇,∇∈LC(M) } . Thesolutions toa fewopenproblemsareannounced. 2.Algebroids,ModulsofAlgebroids,AnomalyFunctions The purpose of this section is to introduce basic notions in the algebraic topology of locally flatmanifolds. 2.1. TheAlgebroidsandModules Givenasmoothfiberbundle B→M thesetof smoothsectionsofB isdenotedbyΓ(B). Definition1. An(abstract) real algebra is a realvector spaceA endowedwithabilinearmap A×A→A Definition2. An(abstract) real two-sidedmodule of an (abstract) algebraA is a realvector spaceWwith two bilinearmappings A×W→W, W×A→W 146