Page - 220 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Accordingtoourconstructiononehas the followingfunctor qΨ= qp2◦ψ∗. This functorqΨ is theHessianfunctorof themodel MΨ=[E1,π,M1,D1,p2◦Ψ]. FurtherΨ×ψ isan isomorphismofMΨontoM2. Letusprove thatassertion(2) impliesassertion(1). Byof thedefinitionofmorphismofmodels, thepairΨ×ψ isan isomorphismofM1 ontoM2 if andonly if p2◦Ψ= p1. Hereweset theexplicit formulas. Let∇∈LC(M1). ForallvectorfieldsX,Y inM2wehave X ·(Y · log(p1))−∇XY · log(p1)=X ·(Y · log(p2◦Ψ))−∇XY · log(p2◦Ψ) =ψ∗(X) ·(ψ∗(Y) · log(p2))−ψ∗(∇XY) · log(p2). nowweobserve that ψ∗(∇XY)= [ψ∗(∇)][ψ∗(X)]ψ∗(Y). Therefore (2) impies theequality ψ∗[q[p2](ψ∗(∇))]= q[p2◦Ψ](∇)= qp1. Thisshowsthe implication (2)→ (1). Letusprove thatassertion(1) impliesassertion(2). Nowweassumethat that (1)holds,viz q[p2◦Ψ] = qp1. Then bothM1 andMΨ have the sameHessian functor. By the virtue of Lemma8 abovewe deduce that p2◦Ψ= p1. Thisends thedemonstration. Reminder. (i) Objects ofGM(Γ,Ξ)arequintuplets M=[E,π,M,D,p]. Theyare called statisticalmodels for themeasurable set (Ξ,Ω). (ii) Objects ofFB(MSE)are functors [E,p]→ [M,D]. Theyare calledMSE-fibrations. 220