Page - 369 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 98 where theconstant cm isgivenby cm= 1m! ×ωm × 8 m(m−1) 4 ,ωm= π m2/2 Γm(m/2) andΓm is themultivariate gammafunctiongiven in [29]. SeeAppendixAfor thederivationofEquation(16) fromEquation(14). 3.RiemannianLaplaceDistributiononPm 3.1.DefinitionofL(Y¯,σ) TheRiemannianLaplacedistributiononPm isdefinedbyanalogywith thewell-knownLaplace distributiononR. Recall thedensityof theLaplacedistributiononR, p(x|x¯,σ)= 1 2σ e−|x−x¯|/σ where x¯∈Randσ>0. This isadensitywithrespect to the lengthelementdxonR. Thedensityof theRiemannianLaplacedistributiononPmwillbegivenby: p(Y|Y¯,σ)= 1 ζm(σ) exp [ −d(Y,Y¯) σ ] (17) here, Y¯∈Pm,σ>0,andthedensity iswithrespect to theRiemannianvolumeelementEquation(13) onPm. Thenormalizingfactorζm(σ)appearing inEquation(17) isgivenbythe integral: ∫ Pm exp [ −d(Y,Y¯) σ ] dv(Y) Assume for now that this integral is finite for some choice of Y¯ and σ. It is possible to show that its valuedoesnotdependon Y¯. Todo so, recall that the actionofGL(m)onPm is transitive. Asaconsequence, thereexistsA∈Pm, such that Y¯= I.A,where I.A isdefinedas inEquation (9). FromEquation(10), it followsthatd(Y,Y¯)= d(Y, I.A)= d(Y.A−1, I). Fromthe invarianceproperty Equation(15): ∫ Pm exp [ −d(Y,Y¯) σ ] dv(Y)= ∫ Pm exp [ −d(Y, I) σ ] dv(Y) (18) The integralontherightdoesnotdependon Y¯,whichproves theaboveclaim. The last integral representationandformulaEquation(16) leadto the followingexplicit expression: ζm(σ)= cm× ∫ Rm e− |r| σ∏ i<j sinh (|ri−rj| 2 ) dr1 · · ·drm (19) where |r|=(r21+ · · ·+rm2 ) 1 2 and cm is thesameconstantas inEquation(16) (seeAppendixBformore detailsonthederivationofEquation(19)). Adistinctive featureof theRiemannianLaplacedistributiononPm, incomparisonto theLaplace distribution onR is that there exist certain values of σ for which it cannot be defined. This is because the integral Equation (19) diverges for certain values of this parameter. This leads to the followingdefinition. Definition1. Setσm = sup{σ> 0 : ζm(σ)<∞}. Then, for Y¯∈Pm andσ∈ (0,σm), theRiemannian LaplacedistributiononPm,denotedbyL(Y¯,σ), is definedas theprobabilitydistributiononPm,whosedensity with respect todv(Y) isgivenbyEquation (17),whereζm(σ) isdefinedbyEquation (19). Theconstantσm in thisdefinitionsatisfies0<σm<∞ forallmandtakes thevalue √ 2 form=2 (seeAppendixCforproofs). 369