Page - 217 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (4) thecategoryF(LC,BF)whoseobjectsare functors GM→BF. Definition60. TheHessian functor is the functor GM M=[E,π,M,D,p]→ qM∈F(LC,BL) HereqM isdefinedby qM[∇]=∇2log(p). Reminder. Werecall that forvectorfieldsX,Ythebilinear formqM[∇](X,Y) isdefinedby qM[∇](X,Y)=X ·(Y · log(p))−∇XY · log(p) ThefunctorqM is calledtheHessianfunctorof themodel M=[E,π,M,D,p]. Ouraimistodemonstratethefollowingclaim.UptoisomorphismastatisticalmodelM isdefined byitsHessianfunctorqM. The functorqM isansignificantcontributionto the informationgeometry. We fix an object ofFB(Γ,Ξ), namely [E,π,M,D]. LetP(E) be the convex set of probability densities in [E,π,M,D]. Themultiplicativegroupofpositive real valued functionsdefined inΞ is denotedbyRΞ+. ThequotientofP(E)moduloRΞ+ isdenotedby PRO(E)=P(E) RΞ+ . Lemma6. Forevery p∈P(E) the imageof M=[E,p], namelyqMdependsonlyon the class [p]∈PRO(E) Proof. Weconsider M=[E,π,M,D,p], M ∗=[E,π,M,D,p∗]. Weassumethat qM= qM∗. Thus inevery local trivializationΘU×Ξonehas the identity X(Ylog( p∗(x,ξ) p(x,ξ) )−∇XYlog(p ∗(x,ξ) p(x,ξ) ))=0 forallX,Y∈Γ(TΘ), forall∇∈LC(Θ). That identityholds ifandonly if the function (x,ξ)→ p ∗(x,ξ) p(x,ξ) belongs toRΞ+. Thisends the idea. 217