Page - 207 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 supports anonnegative realvalued functionPU subject to the followingrequirements. [ρ1.1]: For everyfixedξ∈Ξ the function ΘU θ→PU(θ,ξ) isdifferentiable. [ρ1.2]: For everyfixedθ∈ΘU the triple (Ξ,Ω,PU(θ,−)) is aprobability space. Further the operationof integration ∫ Ξ commuteswith the operationofdifferentiation dθ = ddθ . [ρ1.3]: Let (U,ΦU×φU,PU)and (U∗,ΦU∗×φU∗,PU∗)beas in [ρ1.1]and in [ρ1.2]. IfU∩U∗ =∅ thenPU,PU∗ andγUU∗ are relatedby the formula PU∗ ◦γUU∗=PU. [ρ1.4]: LetU⊂ Mbe an open subset and let γ ∈ Γ. Let us assume that bothU and γ ·Uare domains of local charts (U,ΦU×φU,PU) and (γ ·U,Φγ·U×φγ·U,Pγ·U). Weassumethat those local charts satisfyρ1.1,ρ1.2 andρ1.3. Then the relations Φγ·U◦γ=γ◦ΦU, φγ·U◦γ=γ◦φU, implies the equality Pγ·U◦γ=PU· AComment. Actually, ([ρ1.3])has the followingmeaning: PU∗[γ˜UU∗ ·θU(e),γUU∗ ·ξU(e)]=PU(θU(e),ξU(e)) ∀e∈EU∩U∗. TheFigure4represents (ρ1.3) E∗E E∗E M M∗ M∗M π∗ γ π γ Φ γ Φ φ φ Figure4.Moduli. This ends the comment. 207