Page - 219 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Tosimlplifyweuse the followingnotation. qpj = qMj. In thecategoryFB(Γ,Ξ)weconsideran isomorphism [E1×M1]∈ (e,x)→ [Ψ(e),ψ(x)]∈E2×M2. (1) Letψ∗bethedifferentialofψ. For∇∈LC(M1) the imageψ∗(∇)∈LC(M2) isdefinedby [ψ∗(∇)]X∗Y∗=ψ∗[∇ψ−1∗ (X∗)ψ −1∗ (Y∗)] forallvectorfieldsX∗,Y∗ ∈X(M2). (2) It is clear that thedatum [E1,π,M1,D1,p2◦Ψ] isanobjectof thecategoryGM(X,Ω). Thenfor vectorfieldsX,Y inM1wecalculate (atX,Y) therighthandmemberof the followingequality [qp2◦Ψ(∇)]=∇2[log(p2◦Ψ)]. Direct calculationsyield ∇2[log(p2◦Ψ)](X,Y)=X · [Y · log(p2◦Ψ)]−∇XY · log(p2◦Ψ) =X · [Y · log(p2)◦Ψ]−∇XY · [log(p2)◦Ψ] =ψ∗(X) · [ψ∗(Y) · log(p2)]−ψ∗(∇XY) · log(p2) = [ψ∗(∇)2log(p2)](ψ∗(X),ψ∗(Y)). Thus forall∇∈LC(M1)wehave q[p2◦Ψ](∇)= qp2(ψ∗(∇)). Wesummarize thecalculations just carriedoutas it follows Lemma8. Keeping thenotationwe justusednamely p2 andΨ×ψwehave the followingequality q[p2◦Ψ] = qp2◦ψ∗ Weare inpositionto face theproblemofmoduli space in thecategoryGM(Ξ,Ω). Theorem20. Weconsider twom-dimensional statisticalmodels Mj=[Ej,πj,Mj,Dj,pj], j :=1,2. In the categoryFB(Γ,Ξ) letΨ×ψbean isomorphismof [E1,π1,M1,D1]onto [E2,π2,M2,D2]. The followingassertionsare equivalent. (1) qp2◦ψ∗= qp1, (2) p2◦Ψ= p1. Demonstration. Thedemonstration isbasedonLemmas7and8. 219