Page - 221 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Those categories are canonically equivalent. Further theactions thegroupGin those categories lead to the samemoduli space m= GM(Ξ,Ω) G = FB(MSE) G . Werephrase the theoremonmoduli space. Theorem21. The functor [E,p]→ qp∈BF(M) parametrizes themoduli spacem. Thisends thechallenge2. 10.TheHomologicalStatisticalModels In this section we introduce the theory of homological statistical models (HSM in short). Weaddress the linksbetweenthis theoryandthe local theoryas in [17]. The theory of homological statistical models is useful for strengthening the central role played by the theory of KV homology in the information geometry and in the topology of the information[14,16–18,22,30,37,60,61]. We introduce the theoryof localizationofhomologicalmodels. Weuse it tohighlight the role playedbylocal cohomologicalvanishingtheoremsaswellas theroleplayedbyglobalcohomological vanishingtheorems. Theframework is thecategoryFB(Γ,Ξ). Let [E,π,M,D]beanm-dimensionalobjectof thecategoryFB(Γ,Ξ), vizm= dim(M). TheKV algebra of (M,D) is denoted byA. The smoothmanifoldRm supports a sheaf ofKValgebras A˜. This sheaf is locally isomorphic toA. ThevectorspaceC∞(Rm) isa leftmoduleof A˜. Theaffineaction ofΓ inRm is A˜-preserving. Let (U,ΦU×φU) be a local chart of [E,π,M,D]. We recall that dφU is the differential of φU. Wehave dφU(A)= A˜(φU(U)). Definition61. Ahomologicalmodel consists of the following data. The datum [E,π,M,D] is an object of the categoryFB(Γ,Ξ). Every x ∈Mhas an openneighborhoodUwhich is the domain of a local chart of [E,π,M,D], namely (ΦU×φU).Weset ΘU×Ξ=ΦU(EU). Thosedataare subject to the followingrequirements. HSM.1 :Θ×Ξ supports anonnegative randomsymmetric2-cocycle ΘU×Ξ (θ,ξ)→QU(θ,ξ)∈Z2KV(A˜,R). HSM.2 :Let [U,ΦU×φU,QU]and [U∗,ΦU∗×φU∗,QU∗]as inHSM.1. Ifweassumethat U∩U∗ =∅ then there existsγUU∗ ∈Γ such that HSM.2.1 ΦU∗(e)=γUU∗ ·ΦU(e) ∀e∈EU∩U∗, HSM.2.2 QU(ΦU(e))=γ∗UU∗ · [QU∗(ΦU∗(e))]. 221