Page - 208 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Definition 49. Adatum [U,ΦU×φU,PU,γUU∗] as in the last definition is called a local statistical chart of [E,π,M,D]. Figure4 is representswhatare crucial steps toward theserachof characteristic invariants, viz invariants encoding the points of themoduli space of statisticalmodels. At the present Figure 4 describes themoduli spaceof thecategoryFB(Γ,Ξ) Before dealing with morphisms of the category GM(Ξ,Ω)we introduce a relevant global geometrical invariant. 8.3.2. TheGlobalProbabilityDensityofaStatisticalModel WeconsideraCOMPLETE(ormaximalstatistical)atlasofanobject [E,π,M,D] (of thecategory GM(Ξ,Ω)),namely AΦ=[Uj,Φj,φj,Pj,γij]. ThefamilyUj isanopencoveringofM. ThepairEj×Uj is thedomainof the localchart(Φj×φj). Wehave Ej=EUj. IfUi∩Uj =∅ thenonehas φj(x)= γ˜ji ·φij(x) ∀x∈Ui∩Uj. InparticularA=(Uj,φj) isanaffineatlasof the locallyflatmanifold (M,D).Wehave Φj(Ey∗)=φj(y∗)×Ξ ∀y∗ ∈Uj. Thereforeweset [Ey∗,Ωy∗]=Φ−1j [[φj(y∗)×Ξ],Ω]. TheatlasAΦ satisfiesrequirements(ρ1.1),(ρ1.2)and(ρ1.3). InEUj thelocalfunctionpj isdefinedby pj=Pj◦Φj. Wesuppose that Ui∩Uj =∅. By thevirtueofof [ρ1.3]onehas pi(e)= pj(e) forall e∈EUi∩Uj. Therebythereexistsauniquefunction E e→ p(e)∈R whoserestrictiontoEj coincideswith pj. TherestrictiontoEx isdenotedby px. The triple (Ex,Ωx,px) isaprobabilityspace. Definition50. The function E e→ p(e)∈R is called theprobabilitydensityof themodel [E,π,M,D]. 208