Identifiability of parameters in latent structure models with many observed variables

Annals of Statistics

Volumn 37, Issue 6 A, 2009, Pages 3099-3132

  Allman, Elizabeth S ^a   Matias, Catherine ^a   Rhodes, John A ^b  

^a UNIVERSITY OF ALASKA FAIRBANKS   (United States)

^b University of Paris 5   (France)

Author keywords

Algebraic statistics; Conditional independence; Contingency table; Finite mixture; Identiflability; Latent structure; Multivariate Bernoulli mixture; Nonparametric mixture

Indexed keywords

EID: 69949186291     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOS689     Document Type: Article

References (44)

1. Abo, H., Ottaviani, G. and Peterson, C. (2009). Induction for secant varieties of Segre varieties. Trans. Amer. Math. Soc. 361 767-792. Available at arXiv:math.AG/0607191.MR2452824
2. Allman, E. S. and Rhodes, J. A. (2009). The identifiability of tree topology for phyloge- netic models, including covarion and mixture models. J. Comput. Biol. 13 1101-1113.
3. Allman, E. S. and Rhodes, J. A. (2009). The identifiability of covarion models in phylogenetics. IEEE/ACM Trans. Comput. Biol. Bioinformatics 6 76-88. MR2359648
4. Benaglia, T., Chauveau, D. and Hunter, D. R. (2009). An EM-like algorithm for semi and nonparametric estimation in multivariate mixtures. J. Comput. Graph. Statist. To appear.
5. Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. pringer, New York. MR2159833
6. Carreira-Perpiñán, M. and Renals, S. (2000). Practical identifiability of finite mixtures of multivariate Bernoulli distributions. Neural Comput. 12 141-152.
7. Catalisano, M. V., Geramita, A. V. and Gimigliano, A. (2002). Ranks of tensors, secant varieties of Segre varieties and fat points. Linear Algebra Appl. 355 263-285. MR1930149
8. Catalisano, M. V., Geramita, A. V. and Gimigliano, A. (2005). Higher secant varieties of the Segre varieties double strok P sign;1 × ⋯ × double strok P sign1. J. Pure Appl. Algebra 201 367-380. MR2158764
9. Chambaz, A. and Matias, C. (2009). Number of hidden states and memory: A joint order estimation problem for Markov chains with Markov regime. ESAIM Probab. Stat. 13 38-50.
10. Cox, D., Little, J. and O'Shea, D. (1997). Ideals, Varieties, and Algorithms, 2nd ed. springer, New York. MR1417938
11. Cruz-Medina, I. R., Hettmansperger, T. P. and Thomas, H. (2004). Semiparametric mixture models and repeated measures: The multinomial cut point model. J. Roy. Statist. Soc.Ser. C 53 463-474. MR2075502
12. Daudin, J.-J., Picard, F. andRobin, S. (2008). A mixture model for random graphs. Statist. Comput. 18 173-183.
13. Drton, M. (2006). Algebraic techniques for Gaussian models. In Prague Stochastics (M. Huskova and M. Janzura, eds.) 81-90. Matfyz Press, Prague.
14. Elmore, R., Hall, P. and Neeman, A. (2005). An application of classical invariant theory to identifiability in nonparametric mixtures. Ann. Inst. Fourier (Grenoble) 55 1-28. MR2141286
15. Elmore, R. T., Hettmansperger, T. P. and Thomas, H. (2004) Estimating component cumulative distribution functions in finite mixture models. Comm. Statist. Theory Methods 33 2075-2086. MR2103063
16. Ephraim, Y. and Merhav, N. (2002) Hidden Markov processes. IEEETrans. Inform. Theory 48 1518-1569. MR1909472
17. Finesso, L. (1991) Consistent estimation of the order for Markov and hidden Markov chains. Ph.D. thesis, Univ. Maryland
18. Frank, O. and Harary, F. (1982) Cluster inference by using transitivity indices in empirical graphs. J. Amer. Statist. Assoc. 77 835-840. MR0686407
19. Garcia, L.D., Stillman, M. and Sturmfels, B. (2005) Algebraic geometry of Bayesian networks. J. Symbolic Comput. 39 331-355. MR2168286
20. Gilbert, E. J. (1959) On the identifiability problem for functions of finite Markov chains. Ann. Math. Statist. 30 688-697. MR0107304
21. Glick, N. (1973) Sample-based multinomial classification. Biometrics 29 241-256. MR0347009
22. Goodman, L. A. (1974) Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61 215-231. MR0370936
23. Gyllenberg, M., Koski, T., Reilink, E. and Verlaan, M. (1994) Nonuniqueness in probabilistic numerical identification of bacteria. J. Appl. Probab. 31 542-548. MR1274807
24. Hall, P., Neeman, A., Pakyari, R. and Elmore, R. (2005) Nonparametric inference in multivariate mixtures. Biometrika 92 667-678. MR2202653
25. Hall, P. and Zhou, X.-H. (2003) Nonparametric estimation of component distributions in a ultivariate mixture. Ann. Statist. 31 201-224. MR1962504
26. Hettmansperger, T. P. and Thomas, H. (2000) Almost nonparametric inference for repeated measures in mixture models. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 811-825. MR1796294
27. Koopmans, T.C. and REIERS0L, O. (1950) The identification of structural characteristics. Ann. Math. Statist. 21 165-181. MR0039967
28. Koopmans, T.C., ED. (1950) Statistical Inference in Dynamic Economic Models. Wiley, New York. MR0038640
29. Kruskal, J. B. (1976) More factors than subjects, tests and treatments: An indeterminacy heorem for canonical decomposition and individual differences scaling. Psychometrika 1 281-293. MR0488592
30. Kruskal, J. B. (1977) Three-way arrays: Rank and uniqueness of trilinear decompositions, ith application to arithmetic complexity and statistics. Linear Algebra Appl. 18 95-138. MR0444690
31. Lauritzen, S. L. (1996) Graphical Models Oxford Statistical Science Series 17. Clarendon Press, New York. MR1419991
32. Leroux, B. G. (1992) Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127-143. MR1145463
33. LINDSAY, B. G. (1995) Mixture Models: Theory, Geometry and Applications. NSF-CBMS Regional Conference Series in Probability and Statistics 5.IMS, Hayward, CA.
34. McLachlan, G. and Peel, D. (2000) Finite Mixture Models. Wiley, New York. MR1789474
35. Nigam, K., McCallum, A. K., Thrun, S. and Mitchell, T. M. (2000) Text classification from labeled and unlabeled documents using EM. Machine Learning 39 103-134.
36. Nowicki, K. and SNIJDERS, T. A. B. (2001) Estimation and prediction for stochastic block- structures. J. Amer. Statist. Assoc. 96 1077-1087. MR1947255
37. Pachter, L. and Sturmfels, B. (2005) Algebraic Statistics for Computational Biology. Cambridge Univ. Press, Cambridge. MR2205865
38. Paz, A. (1971) Introduction to Probabilistic Automata. Academic Press, New York. MR0289222
39. Petrie, T. (1969)Probabilistic functions of finite state Markov chains. Ann. Math. Statist. 40 97-115. MR0239662
40. Rothenberg, T. J. (1971) Identification in parametric models. Econometrica 39 577-591. MR0436944
41. Tallberg, C. (2005) A Bayesian approach to modeling stochastic blockstructures with covariates. J. Math. Soc. 29 1-23.
42. Teicher, H. (1967) Identifiability of mixtures of product measures. Ann. Math. Statist. 38 1300-1302. MR0216635
43. Yakowitz, S. J. and Spragins, J. D. (1968) On the identifiability of finite mixtures. Ann. Math. Statist. 39 209-214. MR0224204
44. ZANGHI, H., AMBROISE, C. and MIELE, V. (2008 Fast online graph clustering via Erdös- Rényi mixture. Pattern Recognition 41 3592-3599.