Page - 97 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure14.GeodesicsShootingbetweentwomultivariateGaussian incasen=2. 9. SouriauRiemannianMetric forMultivariateGaussianDensities To illustrate the Souriau-Fishermetric, wewill consider the family ofmultivariateGaussian densitiesandwilldevelopsomeelements thatwehavepreviouslydevelopedpurely theoretically. For the families ofmultivariateGaussiandensities, thatwehave identified as homogeneous manifoldwith theassociated sub-groupof theaffinegroup [ R1/2 m 0 1 ] ,wehave seen that ifwe consider themaselementsofexponential families,wecanwrite ξˆ (elementof thedualLiealgebra) thatplay the roleofgeometricheatQ inSouriauLiegroupthermodynamics, andβ thegeometric (Planck) temperature. ξˆ= [ E [z] E [ zzT ] ] = [ m R+mmT ] , β= ⎡⎢⎣ −R−1m1 2 R−1 ⎤⎥⎦ (245) These elements are homeomorphic to the matrix elements in matrix Lie algebra and dual Liealgebra: ξˆ= [ R+mmT m 0 0 ] ∈ g∗ , β= ⎡⎢⎣ 12R−1 −R−1m 0 0 ⎤⎥⎦ ∈ g (246) IfweconsiderM= [ R′1/2 m′ 0 1 ] , thenwecancompute theco-adjointoperator: Ad∗Mξˆ= [ R+mmT−mm′T R′1/2m 0 0 ] (247) Wecanalsocompute theadjointoperator: AdMβ=M ·β ·M−1= [ R′1/2 m′ 0 1 ]⎡⎣ 12R−1 −R−1m 0 0 ⎤⎦[ R′−1/2 −R′−1/2m′ 0 1 ] AdMβ= ⎡⎣ 12R′1/2R−1R′−1/2 −12R′1/2R−1R′−1/2m′−R′1/2R−1m 0 0 ⎤⎦ (248) 97