Page - 215 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 such ∫ F ω∈Ωq(M). Thevectorspaceof randomdifferentialq-forms isdenotedbyΩq(E). LetΦU×φU bea local chartofM.Weset (ΘU,Ξ)=(φU(U),Ξ)=ΦU(EU). Werecal that inΦU(EU) thepartialdifferentiation ∂∂θ is called thehorizontaldifferentiation inEU. Thereforeweuse therelation ∫ F ◦ ∂ ∂θ = d dθ ◦ ∫ F forsetting thedeRhamcomplexof randomdifferential forms.Namely Ω(E) : 0→R→Ω0(E)→ ...Ωq(E)→Ωq+1(E)...→Ωm(E)→0. ThecomplexΩ(E) isacomplexofΓ-modules.Here Γ=Aut(Ξ,Ω). ThenthecohomologyspaceH∗(Γ,Ω(E)) isbigraded, Hp,q(Γ,Ω(E))=Hp(Γ,Ωq(E)). Theprobabilitydensity p isΓ-invariant. It isanelementofH0,0(Γ,Ω(E)). 8.4.5.AnotherHomologicalNatureofEntropy One ofmain purpose of [14] is the homological nature of the entropy. The classical entropy functionofastatisticalmodel [E,π,M,D,p] isdefinedby E(π(e))= ∫ Eπ(e) p(e∗)log(p(e∗)). In thecomplexΩ(E)weperformthemachineryofEilenberg [59]. Thatyields theexact sequence (of randomcohomologyspaces) →Hq−1res (E,R)→Hqe(E,R)→HqdR(E,R)→H q res(E,R)→ Wetake intoaccount the identities p(γ ·e)= p(e), γ ·(π(e))=π(γ ·e). Thenwehave E(γ ·π(e))= ∫ Eγ·π(e) p(γ ·e∗)log(p(γ ·e∗)) = ∫ Eπ(γ·e) p(e∗)log(p(e∗)) =E(π(e)). Thus theentropyE(π(e)) isΓ-equivariant. Therefore, itdefinesanequivariantcohomologyclass [E]∈H0e(M,R). 215