Page - 218 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 AComment. Themapping M→ qM is a global geometrical invariant in the sense of Erlangen. In otherwords it is an invariant of the Γ-geometry in [E,π,M,D,p]. Ouraimis todemonstrate that M→ qM isacharacteristic invariant in thecategoryGM(Ξ,Ω). Inotherwords the isomorphismclassof the model M=[E,π,(M,D),p] isencodedbythe functor ∇→ qM[∇]. Thefirst step is the following lemma. Lemma7. In the sameobject [E,π,M,D]weconsider twostatisticalmodels M1=[E,π,M,D,p1], M2=[E,π,M,D,p2]. The followingassertionsare equivalent (1) qM1 = qM2, (2) p1= p2. Proof. Wework in thedomainofa local trivializationof [E,π,M,D]. By thevirtueofLemma6above weknowthat qp1 = qp2 if andonly if p1(x,ξ)=λ(ξ)p2(x,ξ) withλ∈RΞ+. Sinceboth p1 and p2 areΓ-equivariant the function Ξ ξ→λ(ξ) isΓ-invariant too.Nowwetakeintoaccount that thenaturalactionofΓ inΞ is transitive. Thereforethe Γ-equivariant functionλ(ξ) isaconstant function. Therefore p1(x,ξ)=λp2(x,ξ) Theoperationof integrationalongafiberofπyields λ=1 Thisends theproof. Weconsider twom-dimensional statisticalmodels for (Ξ,Ω), namely Mj=[Ej,πj,Mj,Dj,pj], j :=1,2. 218