Page - 84 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Wecanthenrewritedensitywithcanonicalvariables: pξˆ(ξ)= 1 Ω∗ e−〈ξ,β〉.dξ e−〈ξ,β〉= 1 Z e−〈ξ,β〉with log(Z)=nlog(2π)+ 1 2 logdet(R)+ 1 2 mTR−1m ξ= [ z zzT ] , ξˆ= [ E [z] E [ zzT ] ] = [ m R+mmT ] , β= [ a H ] = ⎡⎣ −R−1m1 2 R−1 ⎤⎦ with 〈ξ,β〉=Tr[zaT+HTzzT] R=E [ (z−m)(z−m)T ] =E [ zzT−mzT−zmT+mmT]=E[zzT]−mmT (163) Thefirstpotential function(freeenergy/logarithmofcharacteristic function) isgivenby: ψΩ(β)= Ω∗ e−〈ξ,β〉 ·dξ andΦ(β)=−logψΩ(β)= 12 [−Tr[H−1aaT]+ log[(2)ndetH]−nlog(2π)] (164) Weverify therelationbetweenthefirstpotential functionandmoment: ∂Φ(β) ∂β = ∂ [−logψΩ(β)] ∂β = Ω∗ ξ e−〈ξ,β〉 Ω∗ e−〈ξ,β〉 ·dξ ·dξ= Ω∗ ξ ·pξˆ(ξ)·dξ= ξˆ ∂Φ(β) ∂β = ⎡⎢⎣ ∂Φ(β) ∂a ∂Φ(β) ∂H ⎤⎥⎦=[ mR+mmT ] = ξˆ (165) Thesecondpotential function(Shannonentropy) isgivenasaLegendre transformof thefirstone: S(ξˆ)= 〈 ξˆ,β 〉−Φ(β)with ∂Φ(β)∂β = ξˆ and ∂S(ξˆ)∂ξˆ = β S ( ξˆ ) =− Ω∗ e−〈ξ,β〉 Ω∗ e−〈ξ,β〉·dξlog e−〈ξ,β〉 Ω∗ e−〈ξ,β〉·dξ ·dξ=− Ω∗ pξˆ(ξ)logpξˆ(ξ) ·dξ (166) S(ξˆ)=− Ω∗ pξˆ(ξ)logpξˆ(ξ) ·dξ= 1 2 [ log(2)ndet [ H−1 ] +nlog(2π ·e)]= 1 2 [logdet [R]+nlog(2π ·e)] (167) This remarkwasmadeby Jean-Souriau inhisbook [10] as soonas1969. Hehasobserved, as illustrated inFigure10 that ifwetakevectorwith tensorcomponentsξ= ( z z⊗z ) , componentsof ξˆwillprovidemomentsof thefirstandsecondorderof thedensityofprobability pξˆ(ξ).Heusedthis changeofvariablez′=H1/2z+H−1/2a, tocompute the logarithmof thecharacteristic functionΦ(β): 84