Page - 87 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 TheFisher informationdefinesametricgiventoNn aRiemannianmanifoldstructure. The inner productof twotangentvectors (η1,Ω1)∈Tn, (η2,Ω2)∈Tn at thepoint (μ,Σ)∈Nn isgivenby: g(μ,Σ)) ((η1,Ω1) ,(η1,Ω1))=η T 1Σ −1η2+ 1 2 Tr ( Σ−1Ω1Σ−1Ω2 ) (180) NielsChristianBangJespersonhasprovedthat the transformationmodelonRnwithparameter set Rn×Sym+n are exactly those of the form pμ,Σ = fμ,Σλ where λ is the Lebesque measure, where fμ,Σ(x)= h ( (x−μ)TΣ−1(x−μ) ) /det(Σ)1/2 and h : [0,+∞[→R+ isacontinuousfunction with +∞ 0 h(s)s n 2−1ds<+∞. Distributionswithdensitiesof this formarecalledellipticdistributions. To improve understanding of tools, wewill considerGA(n) as a sub-group of affine group, thatcouldbedefinedbyamatrixLiegroupGaf f , thatacts formultivariateGaussianlaws,as illustrated inFigure11: [ 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) (181) Wecanverify thatM is aLiegroupwith classicalproperties, thatproductofMpreserves the structure, theassociativity, thenon-commutativity,andtheexistenceofneutralelement: M1 ·M2= [ R1/21 m1 0 1 ][ R1/22 m2 0 1 ] = [ R1/21 R 1/2 2 R 1/2 1 m2+m1 0 1 ] M2 ·M1= [ R1/22 m2 0 1 ][ R1/21 m1 0 1 ] = [ R1/22 R 1/2 1 R 1/2 2 m1+m2 0 1 ] ⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭ ⇒ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ M1 ·M2∈Gaf f M2 ·M1∈Gaf f M1 ·M2 =M2 ·M1 M1 ·(M2 ·M3)= (M1 ·M2) ·M3 M1 · I=M1 (182) Wecanalsoobserve that the inversepreserves thestructure: M= [ R1/2 m 0 1 ] ⇒M−1R =M−1L =M−1= [ R−1/2 −R−1/2m 0 1 ] ∈Gaf f (183) To thisLiegroupwecanassociateaLiealgebrawhoseunderlyingvector space is the tangent spaceof theLiegroupat the identityelementandwhichcompletelycaptures the local structureof thegroup. ThisLiegroupactssmoothlyonthemanifold,andactsonthevectorfields.Anytangent vectorat the identityofaLiegroupcanbeextendedtoa left (respectivelyright) invariantvectorfield by left (respectivelyright) translating the tangentvector tootherpointsof themanifold. This identifies the tangentspaceat the identityg=TI(G)withthespaceof left invariantvectorfields,andtherefore makes the tangentspaceat the identity intoaLiealgebra, calledtheLiealgebraofG. LG : { Gaf f →Gaf f M →LMN=M ·N andRG : { Gaf f →Gaf f M →RMN=N ·M (184) 87