Page - 222 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Comments. Theequality QU(ΦU(e))=γ∗UU∗ · [QU∗(ΦU∗(e))] has the followingmeaning. Forv,w∈TφU(e)ΘU onehas QU[θ(e),ξ(e)](v,w)=QU∗[γUU∗ ·(θ(e),ξ(e))](d[γUU∗] ·v,d[γUU∗] ·w). Morphismsofhomologicalmodelsaredefinedbyreplacing theprobabilityPU bytherandom cocycleQU. Definition 62. The category of homological statistical models for a measurable set (Ξ,Ω) is denoted by HSM(Ξ,Ω). 10.1. TheCohomologyMappingofHSM(Ξ,Ω) Weconsideranm-dimensionalobjectofHSM(Ξ,Ω)which isdefinedbyacompleteatlas A=[Uj,Φj×φj,γij,Qj] Theunderlyingobjectof theatlasA isdenotedby [E,π,M,D].Weset Θj=φj(Uj)⊂Rm. Wearenotmakinganydifferencebetween (Uj,A)and (Θj,A˜).Weputset Eij=EUi∩Uj. Ifweassumethat Ui∩Uj =∅ thenwehave Φj(e)=γij ·Φi(e), ∀e∈Eij and Qi(Φi(e))=γ∗ij ·Qj(Φj(e)) ∀e∈Eij. Weput qj(e)=Qj(Φj(e)) ∀e∈Ej. Here Ej=EUj. If Ui∩Uj =∅ thenweknowthat [Φj◦Φ−1i ](θi(e),ξi(e))=γij(θi(e),ξi(e)) ∀e∈Eij. Thereforeweget qi(e)= qj(e) ∀e∈Eij. Thereforeqj is therestrictiontoEjofa (globallydefined)map E e→Q(e)∈Z2KV(A,R). 222