An application of classical invariant theory to identifiability in nonparametric mixtures
[Une application de la théorie classique des invariants dans les mélanges non-paramétriques]
Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 1-28.

On sait que l'identifiabilité des mélanges multivariés se réduit à une question de géométrie algébrique. Nous résolvons cette question en étudiant des générateurs particuliers dans l'anneau des polynômes à variables vectorielles, invariants sous l'action du groupe symétrique.

It is known that the identifiability of multivariate mixtures reduces to a question in algebraic geometry. We solve the question by studying certain generators in the ring of polynomials in vector variables, invariant under the action of the symmetric group.

DOI : 10.5802/aif.2087
Classification : 13A50, 62G07, 62H12
Keywords: Mixture model, birational, invariant
Mot clés : modèle de mélange, birationel, invariant
Elmore, Ryan 1 ; Hall, Peter  ; Neeman, Amnon 

1 Australian National University, centre for Mathematics and its Applications, Mathematical Sciences Institute, John Dedman Building, Canberra ACT 0200 (AUSTRALIE)
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Elmore, Ryan; Hall, Peter; Neeman, Amnon. An application of classical invariant theory to identifiability in nonparametric mixtures. Annales de l'Institut Fourier, Tome 55 (2005) no. 1, pp. 1-28. doi : 10.5802/aif.2087. http://www.numdam.org/articles/10.5802/aif.2087/

[1] M. V. Catalisano; A. V. Geramita; A. Gimigliano Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl., Volume 355 (2002), pp. 263-285 | DOI | MR | Zbl

[2] M. V. Catalisano; A. V. Geramita; A. Gimigliano Erratum to ``Ranks of tensors, secant varieties of Segre varieties and fat points'', Linear Algebra Appl., Volume 367 (2003), pp. 347-348 | DOI | MR | Zbl

[3] L. D. Garcia; M. Stillman; B. Sturmfels Algebraic geometry of Bayesian networks (e-print, http://arXiv.org/abs/math.AG/0301255)

[4] I. M. Gel'fand; M. M. Kapranov; A. V. Zelevinsky Discriminants, resultants, and multidimensional determinants, Mathematics: Theory \& Applications, Birkhäuser, Boston, MA, 1994 | MR | Zbl

[5] L. A. Goodman Exploratory latent structure analysis using both identifiable and unidentifiable models, Biometrika, Volume 61 (1974), pp. 215-231 | DOI | MR | Zbl

[6] P. Hall; X.-H. Zhou Nonparametric estimation of component distributions in a multivariate mixture, Ann. Statist., Volume 31 (2003), pp. 201-224 | DOI | MR | Zbl

[7] P. Hall; A. Neeman; R. Pakyari; R. Elmore Nonparametric inference in multivariate mixtures (To appear)

[8] J. M. Landsberg; L. Manivel On the ideals of secant varieties to Segre varieties (e-print, http://arXiv.org/abs/math.AG/0311388) | Zbl

[9] B.G. Lindsay Mixture Models: Theory Geometry and Applications (1995) | Zbl

[10] G.J. Mac; Lachlan; D. Peel Finite Mixture Models, John Wiley & Sons, 2000

[11] A. Mattuck The field of multisymmetric functions, Proc. Amer. Math. Soc., Volume 19 (1968), pp. 764-765 | MR | Zbl

[12] M. Nagata On the normality of the Chow variety of positive 0-cycles of degree m in an algebraic variety (Mem. Coll. Sci. Univ. Kyoto A. Math.), Volume 29 (1955), pp. 165-176 | Zbl

[13] A. Neeman Zero cycles in n , Advances in Math., Volume 89 (1991), pp. 217-227 | DOI | MR | Zbl

[14] E. Netto Vorlesungen über Algebra, Teubner Verlag, Leipzig, 1896 | JFM

[15] H. Teicher Identifiability of mixtures, Ann. Math. Statist., Volume 32 (1961), pp. 244-248 | DOI | MR | Zbl

[16] H. Teicher Identifiability of finite mixtures, Ann. Math. Statist., Volume 34 (1963), pp. 1265-1269 | DOI | MR | Zbl

[17] D.M. Titterington; A.F. Smith; U.E. Makov Statistical Analysis of Finite Mixture Distributions, John Wiley \& Sons, 1985 | MR | Zbl

[18] H. Weyl The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton N.J., 1939 | MR | Zbl

[19] S.J. Yakowitz; J. D. Spragins On the identifiability of finite mixtures, Ann. Math. Statist., Volume 39 (1968), pp. 209-214 | DOI | MR | Zbl

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