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Global identifiability of linear structural equation models. (English) Zbl 1215.62052

Summary: Structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables. We consider the classical case of linear relationships and additive Gaussian noise terms. We give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms. Global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
05C90 Applications of graph theory
62J05 Linear regression; mixed models

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[1] Andrews, D. W. K. and Guggenberger, P. (2010). Asymptotic size and a problem with subsampling and with the m out of n bootstrap. Econometric Theory 26 426-468. · Zbl 1185.62044
[2] Bollen, K. A. (1989). Structural Equations With Latent Variables . Wiley, New York. · Zbl 0731.62159
[3] Brito, C. and Pearl, J. (2002). A new identification condition for recursive models with correlated errors. Struct. Equ. Model. 9 459-474.
[4] Chung, F. R. K. (1997). Spectral Graph Theory. CBMS Regional Conference Series in Mathematics 92 . Amer. Math. Soc., Providence, RI. · Zbl 0867.05046
[5] Cox, D., Little, J. and O’Shea, D. (2007). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra , 3rd ed. Springer, New York. · Zbl 1118.13001
[6] Drton, M., Eichler, M. and Richardson, T. S. (2009). Computing maximum likelihood estimates in recursive linear models with correlated errors. J. Mach. Learn. Res. 10 2329-2348. · Zbl 1235.62077
[7] Drton, M. (2009). Likelihood ratio tests and singularities. Ann. Statist. 37 979-1012. · Zbl 1196.62020
[8] Drton, M. and Yu, J. (2010). On a parametrization of positive semidefinite matrices with zeros. SIAM J. Matrix Anal. Appl. 31 2665-2680. · Zbl 1210.15036
[9] McDonald, R. P. (2002). What can we learn from the path equations?: Identifiability, constraints, equivalence. Psychometrika 67 225-249. · Zbl 1297.62236
[10] Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763-765. · Zbl 0261.62043
[11] Pearl, J. (2009). Causality: Models, Reasoning, and Inference , 2nd ed. Cambridge Univ. Press, Cambridge. · Zbl 1188.68291
[12] Richardson, T. and Spirtes, P. (2002). Ancestral graph Markov models. Ann. Statist. 30 962-1030. · Zbl 1033.60008
[13] Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search , 2nd ed. MIT Press, Cambridge, MA. · Zbl 0806.62001
[14] Shpitser, I. and Pearl, J. (2006). Identification of joint interventional distributions in recursive semi-Markovian causal models. In Proceedings of the 21st National Conference on Artificial Intelligence 1219-1226. AAAI Press, Menlo Park, CA.
[15] Tian, J. (2002). Studies in causal reasoning and learning. Ph.D. thesis, Computer Science Dept., Univ. California, Los Angeles.
[16] Tian, J. (2009). Parameter identification in a class of linear structural equation models. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Pasadena, California 1970-1975. Morgan Kaufmann, San Francisco, CA.
[17] Wermuth, N. (2010). Probability distributions with summary graph structure. Bernoulli . To appear. Available at . · Zbl 1245.62062
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