Page - 249 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 383 inducedsubgraphofG. ThegraphG is said tobechordalordecomposable ifGcontainsnocycleof lengthgreater than threeasan inducedsubgraph,andsaid tobeA4-free ifG containsnoA4 graph •−•−•−• as an induced subgraph. Let bea total order on thevertex setVG, and for i∈VG, putV[i]G := { j∈VG ; i∼ jand i j}⊂VG. Then, is said tobeaneliminatingorderon thegraph G if the inducedsubgraphwiththevertexsetV[i]G is complete foreach i∈VG. It isknownthat there existsaneliminatingorderonG if andonly if thegraphG is chordal. Letus identify thevertex setVG with{1,2,. . . ,r}. LetZG be the spaceof symmetricmatrices x= (xij) ∈ Sym(r,R), such that, if i = j and i ∼ j, then xij = 0. DefinePG :=ZG∩Pr. We can show([11] (Theorem2.2), [26]) that theconePG ishomogeneous ifandonly if thegraphG is chordal and A4-free. On the other hand, it is known in the graphicalmodel theory aswell as the sparse matrix linearalgebra thateventhoughPG isnothomogeneous,various formulasstillhold forPG ifG is chordal. TheconePG isexpressedasPVwithn1=n2= ···=nr=1and: Vji= { R (j∼ i), {0} (j ∼ i). Then, the condition (V2)means exactly that the order≤ is an eliminating order onG. Therefore, anyconePGwithchordalGcanbe treatedasPV inour framework.Mostof the integral formulas for PG in [11,27] canbededucedfromTheorems3and4,while theWishartdistribution isacentralobject in the theoryofgraphicalmodel. In [28], theanalysis forgeneralizedpower functionsassociatedwith all eliminatingorders isdiscussedforaspecificgraphAn :•−•−···−•bydirect computations. Acknowledgments: Theauthorwould like toexpresssinceregratitudetoPiotrGraczykandYoshihikoKonno forstimulatingdiscussionsabout this subject.He isalsograteful toFrédéricBarbaresco forhis interest inand encouragementof thiswork.Hethanks toanonymousreferees forvaluablecomments,whichwerehelpful for the improvementof thepresentpaper. ThisworkwassupportedbyJSPSKAKENHIGrantNumber16K05174. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. References 1. Koszul, J.L.Ouvertsconvexeshomogènesdesespacesaffines.Math.Z. 1962,79, 254–259. 2. Vinberg,E.B.Thetheoryofconvexhomogeneouscones.Trans.MoscowMath. Soc. 1963,12, 340–403. 3. Vey, J.Sur lesautomorphismesaffinesdesouvertsconvexessaillants.AnnalidellaScuolaNormaleSuperiorediPisa 1970,24, 641–665. 4. Shima,H.TheGeometryofHessianStructures;WorldScientific:Hackensack,NJ,USA,2007. 5. Nesterov, Y.; Nemirovskii, A. Interior-Point Polynomial Algorithms in Convex Programming; Society for IndustrialandAppliedMathematics: Philadelphia,PA,USA,1994. 6. Barbaresco, F.Koszul informationgeometry andSouriaugeometric temperature/capacity of Lie group thermodynamics.Entropy2014,16, 4521–4565. 7. Barbaresco, F. Symplectic structure of information geometry: Fisher etric andEuler-Poincaré equation of Souriau Lie group thermodynamics. InGeometric Science of Information; Nielsen, F., Barbaresco, F., Eds.; (LectureNotes inComputer Science); Springer InternationalPublishing: Basel, Switzerland, 2015; Volume9389,pp. 529–540. 8. Gindikin,S.G.Analysis inhomogeneousdomains.Russ.Math. Surv. 1964,19, 1–89. 9. Truong, V.A.; Tunçel, L. Geometry of homogeneous convex cones, duality mapping, and optimal self-concordantbarriers.Math. Program. 2004,100, 295–316. 10. Tunçel, L.; Xu, S.Onhomogeneous convex cones, theCaratheodorynumber, and thedualitymapping. Math.Oper.Res. 2001,26, 234–247. 11. Letac,G.;Massam,H.Wishartdistributions fordecomposablegraphs.Ann. Stat. 2007,35, 1278–1323. 12. Rothaus,O.S.Theconstructionofhomogeneousconvexcones.Ann.Math. 1966,83, 358–376. 13. Xu,Y.-C.TheoryofComplexHomogeneousBoundedDomains;Kluwer:Dordrecht,TheNetherlands,2005. 249