Page - 85 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Figure10. Introductionofpotential functionformultivariateGaussian lawinSouriaubook[10]. Wecanfinallycompute themetric fromthematrixgij: ds2=∑ ij gijdθidθj= dmTR−1dm+ 1 2 Tr [( R−1dR )2] (168) andfromclassicalexpressionof theEuler-Lagrangeequation: n ∑ i=1 gik .. θi+ n ∑ i,j=1 Γijk . θi . θj=0 , k=1,...,nwithΓijk= 1 2 [ ∂gjk ∂θi + ∂gjk ∂θj + ∂gij ∂θk ] (169) That isexplicitelygivenby[170]:{ .. R+ . m . mT− .RR−1 .R=0 .. m− .RR−1 .m=0 (170) Wecannot integrate thisEuler-Lagrangeequation.Wewill see thatLiegrouptheorywillprovide newreducedequation,Euler-Poincaréequation,usingSouriautheorem. Wemakereference to thebookofDeza thatgivesasurveyaboutdistanceandmetric space [171]. ThecaseofNaturalExponential families thatare invariantbyanaffinegrouphasbeenstudiedby Casalis (in1999paperandinherPh.D. thesis) [172–178]andbyLetac [179–181].Wegive thedetails ofCasalis’development inAppendixC.Barndorff-Nielsenhasalsostudiedtransformationmodels for exponential families [182–186]. In this section,wewill only consider the case ofmultivariate Gaussiandensities. 8.AffineGroupActionforMultivariateGaussianDensitiesandSouriau’sMomentMap: ComputationofGeodesicsbyGeodesicShooting Tomore deeply understandKoszul and SouriauLie groupmodels of information geometry, wewill illustrate their tools formultivariateGaussiandensities. Consider thegeneral linear groupGL(n) consistingof the invertiblen×nmatrices, that is a topologicalgroupacting linearlyonRnby: 85