Page - 203 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (1) Ifθ = θ∗ there exists t∗∗ ∈Rm such that P(θ,t∗∗) =P(θ∗,t∗∗), (2) P(θ,t)≤1∀(θ,t)∈TTm, (3) ∫ Rm P(θ,t)dt=1. Wededuce that thepair (Tm,P) is anm-dimensionalmanifoldofprobabilitydensities in themeasurable set (Rm,β(Rm)). The imageof every local chart ofTm isa local statisticalmodel in the classical sense [17,18,22]. This ends theDigression. Wearemotivatedfor introducinganewtheoryofstatisticalmodelswhose localizationyields the current theory. Thetheoryweintroduce isananswer toMcCullaghandtoGromov. 8.1. ThePreliminaries In thisPartBweface threemajorchallenges. Challenge1.Taking intoaccount thecriticismswehaveraisedouraimis to introduceanewtheoryof statisticalmodelwhose localization leads to theclassical theoryofstatisticalmodels. Challenge2.Thesecondchallenge is theSearch foran invariantwhichencodes thepointof themoduli spaceof isomorphismclassofstatisticalmodels. Challenge3.Weintroduce the theoryofhomological statisticalmodelandweexplore the linksbetween this theoryandthechallenge2. Challenge 4. The fourth challenge is to explore the relationshipsbetween“challenge1, challenge2, challenge3”and“VanishingTheoremsin the theoryofKVhomology”. ThetheoryofKVcohomologyandthegeometryofKoszulplay importantroles.Weintroduce theneededdefinitions. Let (Ξ,Ω)beameasurable set. LetAut(Ξ,Ω)be thegroupofmeasurable isomorphismsofΞ. Let (M,D)bea locallyflatmanifoldwhoseKValgebra isdenotedbyA. Wekeep thenotationused inPartAof this paper. For instanceS2(M) is the vector space of differentiablesymmetricbi-linear forms inM. Definition44. ArandomHessianmetric in (M,D) is amap M×Ξ (x,ξ)→Q(x,ξ)∈S2[T∗xM], whichhas the followingproperties (1) foreveryvector fieldXtherealnumberQ(x,ξ)[X,X] isnonnegative, furthermore∀v∈TxM\{0}∃ξ∈Ξ such that Q(x,ξ)(v,v)>0, (2) for everyξ∈Ξ, the randomKVcochain (X,Y)→Qξ(X,Y)(x) with Qξ(X,Y)(x)=Q(x,ξ)(Xx,Yx) is a randomcocycle of theKVcomplex [C∗KV(A,C∞(M)),δKV]. 203