Page - 248 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 Then, thesetHV of lower triangularmatricesTof the form: T= ⎛⎜⎜⎜⎜⎝ T11 T21 T22 ... ... Tr1 Tr2 . . . Trr ⎞⎟⎟⎟⎟⎠ becomes a linear Lie group, andHV acts on the spaceZV by ρ(T)x := TxtT (T ∈ HV, x ∈ ZV). ThegroupHV actsontheconePV simplytransitivelybythisactionρ, so thatPV isahomogeneous cone.Moreover, it is shownin[15] thateveryhomogeneouscone is linearly isomorphic tosuchPV (seealso [18]). LetV0={V0lk}1≤k<l≤3 be thesystemgivenbyV021={0}andV0lk=R ((l,k) =(2,1)). Then: ZV0 = ⎧⎪⎨⎪⎩ ⎛⎜⎝x1 0 x40 x2 x5 x4 x5 x3 ⎞⎟⎠ ; x1,. . . ,x5∈R ⎫⎪⎬⎪⎭ , andPV0 :=ZV0∩P3 ishomogeneousbecause (V1)–(V3)aresatisfiedin thiscase.Ontheotherhand, letV1={V1lk}1≤k<l≤3 be thesystemgivenbyV131={0}andV1lk=R ((l,k) =(3,1)). Then: ZV1 = ⎧⎪⎨⎪⎩ ⎛⎜⎝x1 x4 0x4 x2 x5 0 x5 x3 ⎞⎟⎠ ; x1,. . . ,x5∈R ⎫⎪⎬⎪⎭ . Note thatV1 satisfiesonly(V1)and(V2),butPV1 ishomogeneousbecausePV1 is isomorphic to the homogeneousconePV0 via themap: PV1 ⎛⎜⎝x1 x4 0x4 x2 x5 0 x5 x3 ⎞⎟⎠ → ⎛⎜⎝1 0 00 0 1 0 1 0 ⎞⎟⎠ ⎛⎜⎝x1 x4 0x4 x2 x5 0 x5 x3 ⎞⎟⎠ ⎛⎜⎝1 0 00 0 1 0 1 0 ⎞⎟⎠= ⎛⎜⎝x1 0 x40 x3 x5 x4 x5 x2 ⎞⎟⎠∈PV0. Thisexample tellsus thatourmatrix realizationofaconvexcone isnotuniqueandthat thecondition (V3) ismerelyasufficientconditionfor thehomogeneityof thecone. Many ideas in this work are inspired by the theory of homogeneous cones. The notion of generalized power functions, as well as the Γ-type integral formulas are due to Gindikin [8] (seealso [23]). TheWishartdistributions forhomogeneousconesarestudied in [17,21,24,25]. 7.2. ConesAssociatedwithChordalGraphs Ifn1=n2= ···=nr=1, thenVlkequalseitherRor{0}. Inthiscase,ZV is thespaceofsymmetric matriceswithprescribedzerocomponents. Suchaspace isdescribedbyusinganundirectedgraphin thegraphicalmodel theory. Letus recall somenotion in thegraphtheory. LetGbeagraphandVG thesetofverticesofG. WeassumethatGhasnomultipleedge, that is, foranytwovertices i, j∈VG, either there isoneedge connecting themor there isnoedgebetweenthem.Theserelationsof thevertices iand jaredenoted by i∼ jand i ∼ j, respectively.Assumefurther thatGhasnoloop,whichmeans that i ∼ i for i∈VG. WedefinetheedgesetEG⊂VG×VGby: EG :={(i, j)∈VG×VG ; i∼ j} . SinceVG andEGhaveallof the informationofG, thegraphG isoften identifiedwith thepair (VG,EG). For a non-empty subsetV′ ofVG, putE′ := EG∩(V′×V′). The graphG′ := (V′,E′) is called an 248