Page - 53 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 ⎧⎨⎩ . m=Rb . R=R ( B−bmT) (10) thatwesolveby“geodesicshooting” technicbasedonEriksenequationofexponentialmap. • For the families ofmultivariateGaussiandensities, thatwehave identified as homogeneous manifoldwith theassociatedsub-groupof theaffinegroup [ R1/2 m 0 1 ] ,wehaveconsidered theelementsofexponential families, thatplaytheroleofgeometricheatQ inSouriauLiegroup thermodynamics,andβ thegeometric (Planck) temperature: Q= ξˆ= [ E [z] E [ zzT ] ] = [ m R+mmT ] , β= ⎡⎢⎣ −R−1m1 2 R−1 ⎤⎥⎦ (11) Wehaveconsideredthat theseelementsarehomeomorphto the (n+1)× (n+1)matrixelements: Q= ξˆ= [ R+mmT m 0 0 ] ∈ g∗ , β= ⎡⎢⎣ 12R−1 −R−1m 0 0 ⎤⎥⎦ ∈ g (12) tocompute theSouriausymplecticcocycleof theLiegroup: θ(M)= ξˆ(AdM(β))−Ad∗Mξˆ (13) where theadjointoperator isequal to: AdMβ= ⎡⎣ 12Ω−1 −Ω−1n 0 0 ⎤⎦withΩ=R′1/2RR′−1/2 andn=(1 2 m′+R′1/2m ) (14) with ξˆ(AdM(β))= [ Ω+nnT n 0 0 ] (15) andtheco-adjointoperator: Ad∗Mξˆ= [ R+mmT−mm′T R′1/2m 0 0 ] (16) • Finally,wehave computed theSouriau-Fishermetric gβ([β,Z1] , [β,Z2]) = Θ˜β(Z1, [β,Z2]) for multivariateGaussiandensities,givenby: gβ([β,Z1] , [β,Z2])= Θ˜β(Z1, [β,Z2])= Θ˜(Z1, [β,Z2])+ 〈 ξˆ, [Z1, [β,Z2]] 〉 = 〈Θ(Z1) , [β,Z2]〉+ 〈 ξˆ, [Z1, [β,Z2]] 〉 (17) withelementofLiealgebragivenbyZ= ⎡⎣ 12Ω−1 −Ω−1n 0 0 ⎤⎦. Theplanof thepaper isas follows.After this introduction inSection1,wedevelopinSection2 the position of Souriau symplecticmodel of statistical physics in the historical developments of 53