Page - 145 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Section 6. This section is devoted to the homological version of the geometry of Koszul. Our approach involves the dualistic relation of Amari. The KV cohomology links the dualistic relationwith thegeometryofKoszul. Section7. In this sectionsummarize thehighlightingfeaturesofPartA. PartB:Sections8–14. Section8. This is thestartingsectionof thesecondpartB.ThisPartB isdevotedtonewinsights in the theoryofstatisticalmodels.On2002PeterMcCullar raisedtheprovocativequestion. What IsaStatisticalModel Across theworld (Australia,Canada,Europe,US) theMcCullaghs paperbecametheobjectof manycriticismsandquestionsbyeminent theoreticalandappliedstatisticians [30]. Part B is aimed at supplying some deficiencies in the current theory of statistical models. Weaddresssomecriticismswhichsupport theneedofre-establishingthe theoryofstatisticalmodel formeasurablesets. Thosecriticismsareusedforhighlight the lackofbothStructureandRelations. Thosecriticismsalsohighlight thesearchofM.Gromov[15]. Theneedfor structuresandrelations was the intuitionofPeterMcCullagh. Looselyspeaking there isa lackof IntrinsicGeometry in the senseofErlangen. Subsequently the lackof intrinsicgeometryyieldsother things that are lacking. Theproblemof themoduli spaceofmodels isnotstudied,althoughthiswouldbecrucial forapplied informationgeometry,andforappliedstatistics. Thatmightbeakey inreadingsomethecontroversy about [30]. Section9. Inthissectionweaddresstheproblemofmodulispaceofstatisticalmodels. Theproblem of moduli space in a category is a major question in Mathematic. It is generally a difficult problemthat involvesfindingacharacteristic invariantwhichencodes thepointof themoduli space. Suchaninvariant is a crucial step toward the geometry and the topology of a moduli space. Amongotherneeds, theproblemofencodingthemoduli spaceofmodelshasmotivatedourneedof anewapproach, that is tosaytheneedofa theoryhavingnicemathematical structureandrelations. In thisPartB theproblemof themoduli space is solved. Tosummarize the theoremdescribing the moduli spacesofstatisticalmodelsweneedthe followingnotation. A gauge structure in a manifold M is a pair (M,∇)where∇ is a Koszul connection in M. Thecategoryofgaugestructures inM isdenotedbyLC(M).Weareconcernedwith thevectorbundle T∗⊗2M of bi-linear forms in the tangent bundle TM. The sheaf of sections of T∗⊗2M is denoted byBL(M). Thecategoryofm-dimensionalstatisticalmodels(tobedefined)ofameasurable(Ξ,Ω) isdenoted byGMm(Ξ,Ω). Thecategoryof randomfunctors LC(M)×Ξ→BL(M) is denoted byF(LC,BL)(M). One of the interesting breakthrough in this Part.B is the following solutionto theproblemofmoduli. Theorem1. There exists a functor GMm(Ξ,Ω) M→ qM∈BL(M) (2) whichdeterminesamodelMupto isomorphism. Let pbe theprobabilitydensityof amodelM. Themathematical expectationof qM(∇) isdefinedby E(qM(∇))= ∫ Ξ pqM(∇). (3) ThequantityE(qM)(∇)doesnotdependontheKoszul connection∇. It is called theFisher informationofM. 145