Page - 52 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 • Wehaveestablished that theaffinerepresentationofLiegroupandLiealgebraby Jean-Marie Souriau isequivalent to Jean-LouisKoszul’saffinerepresentationdevelopedin the frameworkof hessiangeometryof convexsharpcones. BothSouriauandKoszulhaveelaboratedequations requestedforLiegroupandLiealgebratoensuretheexistenceofanaffinerepresentation.Wehave comparedbothapproachesofSouriauandKoszul ina table. • Wehave applied the Souriaumodel for exponential families and especially formultivariate Gaussiandensities. • Wehaveapplied theSouriau-KoszulmodelGibbsdensity to compute themaximumentropy density forsymmetricpositivedefinitematrices,usingthe innerproduct〈η,ξ〉=Tr(ηTξ),∀η,ξ∈ Sym(n)givenbyCartan-Killingform.TheGibbsdensity(generalizationofGaussianlawfortheses matricesanddefinedasmaximumentropydensity): pξˆ(ξ)= e −〈Θ−1(ξˆ),ξ〉+Φ(Θ−1(ξˆ)) =ψΩ(Id) · [ det ( αξˆ−1 )] ·e−Tr(αξˆ−1ξ) withα= n+1 2 (5) • For thecaseofmultivariateGaussiandensities,wehaveconsideredGA(n)asub-groupofaffine group, thatwedefinedbya(n+1)× (n+1)embedding inmatrixLiegroupGaf f , andthatacts formultivariateGaussian lawsby: [ Y 1 ] = [ R1/2 m 0 1 ][ X 1 ] = [ R1/2X+m 1 ] , ⎧⎪⎪⎨⎪⎪⎩ (m,R)∈Rn×Sym+(n) M= [ R1/2 m 0 1 ] ∈Gaf f X≈ℵ(0, I)→Y≈ℵ(m,R) (6) • FormultivariateGaussiandensities,aswehave identifiedtheactingsub-groupofaffinegroup M,wehavealsodevelopedthecomputationof theassociatedLiealgebrasηL andηR, adjointand coadjointoperators,andespecially theSouriau“momentmap”ΠR:〈 nL,M−1nRM 〉 = 〈ΠR,nR〉 withM= [ R1/2 m 0 1 ] , nL= ⎡⎣ R−1/2 .R1/2 R−1/2 .m 0 0 ⎤⎦ andηR= ⎡⎣ R−1/2 .R1/2 .m−R−1/2 .R1/2 .m 0 0 ⎤⎦ ⇒ΠR= ⎡⎣ R−1/2 .R1/2+R−1 .mmT R−1 .m 0 0 ⎤⎦ (7) Using Souriau Theorem (geometrization ofNoether theorem), weuse the property that this momentmapΠR is constant (its componentsareequal toNoether invariants): dΠR dt =0⇒ ⎧⎨⎩ R −1 .R+R−1 .mmT=B= cste R−1 .m= b= cste (8) to reduce theEuler-LagrangeequationofgeodesicsbetweentwomultivariateGaussiandensities:⎧⎨⎩ .. R+ . m . mT− .RR−1 .R=0 .. m− .RR−1 .m=0 (9) to this reducedequationofgeodesics: 52