Seite - 82 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 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) (151) Thenwecanrecover therelationwithFishermetric: I(β)=−E [ ∂2logpβ(ξ) ∂β2 ] =−E [ ∂2(−〈ξ,β〉+Φ(β)) ∂β2 ] =−∂ 2Φ(β) ∂β2 ξˆ= ∂Φ(β) ∂β I(β)=E [ ∂logpβ(ξ) ∂β ∂logpβ(ξ) ∂β T] =E [( ξ− ξˆ)(ξ− ξˆ)T]=E[ξ2]−E [ξ]2=Var(ξ) (152) withCrouzeix relationestablished in1977 [147,148], ∂2Φ ∂β2 = [ ∂2S ∂ξˆ2 ]−1 giving thedualmetric, indual space,whereentropySand(minus) logarithmofcharacteristic function,Φ, aredualpotential functions. Thefirstmetricof informationgeometry [149,150], theFishermetric isgivenbythehessianof the characteristic function logarithm: I(β)=−E [ ∂2logpβ(ξ) ∂β2 ] =−∂ 2Φ(β) ∂β2 = ∂2logψΩ(β) ∂β2 (153) ds2g=dβ TI(β)dβ=∑ ij gijdβidβjwithgij=[I(β)]ij (154) Thesecondmetricof informationgeometry isgivenbyhessianof theShannonentropy: ∂2S(ξˆ) ∂ξˆ2 = [ ∂2Φ(β) ∂β2 ]−1 withS(ξˆ)= 〈 ξˆ,β 〉−Φ(β) (155) ds2h=dξˆ T [ ∂2S(ξˆ) ∂ξˆ2 ] dξˆ=∑ ij hijdξˆidξˆjwithhij= [ ∂2S(ξˆ) ∂ξˆ2 ] ij (156) Bothmetricswillprovide thesamedistance: ds2g=ds 2 h (157) FromtheCartan innerproduct,wecangenerate logarithmof theKoszulcharacteristic function, anditsLegendre transformtodefineKoszulentropy,KoszuldensityandKoszulmetric, asexplained in the followingFigure9: 82