Page - 81 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 This characteristic function is at the cornerstoneofmodern concept of informationgeometry, definingKoszuldensitybysolutionofmaximumKoszul-Shannonentropy[140]: Max p [ − Ω∗ pξˆ(ξ)logpξˆ(ξ) ·dξ ] suchthat Ω∗ pξˆ(ξ)dξ=1and Ω∗ ξ ·pξˆ(ξ)dξ= ξˆ (146) pξˆ(ξ)= e−〈Θ−1(ξˆ),ξ〉 Ω∗ e−〈Θ−1(ξˆ),ξ〉.dξ ξˆ=Θ(β)= ∂Φ(β)∂β whereΦ(β)=−logψΩ(β) ψΩ(β)= Ω∗ e−〈β,ξ〉dξ , S(ξˆ)=− Ω∗ pξˆ(ξ)logpξˆ(ξ) ·dξ andβ=Θ−1(ξˆ) S(ξˆ)= 〈 ξˆ,β 〉−Φ(β) (147) This last relation isaLegendre transformbetweenthe logarithmofcharacteristic functionand theentropy: logpξˆ(ξ)=−〈ξ,β〉+Φ(β) S( − ξ)=− Ω∗ pξˆ(ξ) · logpξˆ(ξ) ·dξ=−E [ logpξˆ(ξ) ] S( − ξ)= 〈E [ξ] ,β〉−Φ(β)= 〈ξˆ,β〉−Φ(β) (148) The inversionΘ−1(ξˆ) is given by the Legendre transform based on the property that the Koszul-Shannon entropy is given by the Legendre transform of minus the logarithm of the characteristic function: S(ξˆ)= 〈 β, ξˆ 〉−Φ(β)withΦ(β)=−log Ω∗ e−〈ξ,β〉dξ ∀β∈Ωand∀ξ, ξˆ∈Ω∗ (149) We can observe the fundamental property that E [S(ξ)] = S(E [ξ]) , ξ ∈ Ω∗, and also as observedbyMauriceFréchet that“distinguishedfunctions” (densitieswithestimator reaching the Fréchet-Darmoisbound)aresolutionsoftheAlexisClairautequation introducedbyClairautin1734[141], as illustrated inFigure8: S(ξˆ)= 〈 Θ−1(ξˆ), ξˆ 〉 −Φ [ Θ−1(ξˆ) ] ∀ξˆ∈{Θ(β)/β∈Ω} (150) Figure8.Clairaut-Legendreequation introducedbyMauriceFréchet inhis1943paper [141]. DetailsofFréchetelaborationfor thisClairaut(-Legendre)equationfor“distinguishedfunction” isgiven inAppendixA,andotherelementsareavailableonFréchet’spapers [141–144]. In this structure, theFishermetric I(x)makesappearnaturallyaKoszulhessiangeometry [145,146], ifweobserve that 81