Page - 211 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Definition55. Astatisticalmodel for ameasurable set (Ξ,Ω) is a functor of the categoryFB(Γ,Ξ) in the categoryFB(MSE, namely [E,π,M,D]→ [E,π,M,D,p] Atthepresentstep it is clear that the informationgeometry is structured. 8.3.5. Fisher Information inGM(Ξ,Ω) WeconsideraMSE-fibration M :=[E,p]→ (M,D). TheFisher informationtobedefinedisanelementgofΓ(S2(T∗M)). Werecall thateveryMSE-fiberMx,x∈Mhasastructureofprobabilityspace Mx :=[Ex,Ωx,px]. LetX,Ybe localvectorfieldswhicharedefinedinaopenneighbourhoodofx∈M. Definition56. TheFisher informationat x isdefinedby gx(X,Y)=− ∫ Ex p(e)[D2log(p(e))](X,Y)d(e) We recall that the horizontal differentiation commutes with the integration along the MSE-fibers,viz dθ◦ ∫ F = ∫ F ◦ ∂ ∂θ . So theFisher informationg iswelldefined. Ithas the followingproperties (1) g ispositivesemi-definite, (2) g isan invariantof theΓ-geometry in [E,π,M,D,p]. 8.4. ExponentialModels Let [E,π,M,D,p]beanobjectofGM(Ξ,Ω).Werecall thatdatawhicharedefinedinE arecalled randomdata in thebasemanifoldM. Theoperationof integrationalongtheMSE-fibers isdenoted by ∫ F. Thusarandomdatumμ is calledsmooth if its image ∫ F(μ) is smooth. Converselyeverydatumθ∗which ispoint-wisedefinedinM is the imageof therandomdatum θ= θ∗◦π. Soweget θ∗= ∫ F [θ∗◦π]. Thusateveryx∈Monehas θ∗(x)= ∫ Ex θ∗(π(e))px(e)de. Arandomaffinefunction isa function E e→ a(e)∈R 211