Annals of Statistics

Half-trek criterion for generic identifiability of linear structural equation models

Rina Foygel, Jan Draisma, and Mathias Drton

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Abstract

A linear structural equation model relates random variables of interest and corresponding Gaussian noise terms via a linear equation system. Each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms. We study parameter identifiability in these models, that is, we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated distribution. We treat the case of generic identifiability, where unique recovery is possible for almost every choice of parameters. We give a new graphical condition that is sufficient for generic identifiability and can be verified in time that is polynomial in the size of the graph. It improves criteria from prior work and does not require the directed part of the graph to be acyclic. We also develop a related necessary condition and examine the “gap” between sufficient and necessary conditions through simulations on graphs with $25$ or $50$ nodes, as well as exhaustive algebraic computations for graphs with up to five nodes.

Article information

Source
Ann. Statist., Volume 40, Number 3 (2012), 1682-1713.

Dates
First available in Project Euclid: 2 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1349196388

Digital Object Identifier
doi:10.1214/12-AOS1012

Mathematical Reviews number (MathSciNet)
MR3015040

Zentralblatt MATH identifier
1257.62059

Subjects
Primary: 62H05: Characterization and structure theory 62J05: Linear regression

Keywords
Covariance matrix Gaussian distribution graphical model multivariate normal distribution parameter identification structural equation model

Citation

Foygel, Rina; Draisma, Jan; Drton, Mathias. Half-trek criterion for generic identifiability of linear structural equation models. Ann. Statist. 40 (2012), no. 3, 1682--1713. doi:10.1214/12-AOS1012. https://projecteuclid.org/euclid.aos/1349196388


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References

  • Bollen, K. A. (1989). Structural Equations with Latent Variables. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR996025
    Zentralblatt MATH: 0731.62159
  • Brito, C. (2004). Graphical methods for identification in structural equation models. Ph.D. thesis, UCLA Computer Science Dept.
  • Brito, C. and Pearl, J. (2002a). A new identification condition for recursive models with correlated errors. Struct. Equ. Model. 9 459–474.
    Mathematical Reviews (MathSciNet): MR1930449
    Digital Object Identifier: doi:10.1207/S15328007SEM0904_1
  • Brito, C. and Pearl, J. (2002b). A graphical criterion for the identification of causal effects in linear models. In Proceedings of the Eighteenth National Conference on Artificial Intelligence (AAAI) 533–538. AAAI press, Palo Alto, CA.
  • Brito, C. and Pearl, J. (2006). Graphical condition for identification in recursive SEM. In Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (R. Dechter and T. S. Richardson, eds.) 47–54. AUAI Press, Arlington, VA.
  • Chan, H. and Kuroki, M. (2010). Using descendants as instrumental variables for the identification of direct causal effects in linear SEMs. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (Y. W. Teh and M. Titterington, eds.). J. Mach. Learn. Res. (JMLR), Workshop and Conference Proceedings 9 73–80. Available at http://jmlr.csail.mit.edu/proceedings/.
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2001). Introduction to Algorithms, 2nd ed. MIT Press, Cambridge, MA.
    Mathematical Reviews (MathSciNet): MR1848805
  • Cox, D., Little, J. and O’Shea, D. (2007). Ideals, Varieties, and Algorithms, 3rd ed. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2290010
  • Decker, W., Greuel, G.-M., Pfister, G. and Schönemann, H. (2011). Singular 3-1-3—A computer algebra system for polynomial computations. Available at http://www.singular.uni-kl.de.
  • Didelez, V., Meng, S. and Sheehan, N. A. (2010). Assumptions of IV methods for observational epidemiology. Statist. Sci. 25 22–40.
    Mathematical Reviews (MathSciNet): MR2741813
    Digital Object Identifier: doi:10.1214/09-STS316
    Project Euclid: euclid.ss/1280841731
  • Drton, M., Foygel, R. and Sullivant, S. (2011). Global identifiability of linear structural equation models. Ann. Statist. 39 865–886.
    Mathematical Reviews (MathSciNet): MR2816341
    Zentralblatt MATH: 1215.62052
    Digital Object Identifier: doi:10.1214/10-AOS859
    Project Euclid: euclid.aos/1299680957
  • Evans, W. N. and Ringel, J. S. (1999). Can higher cigarette taxes improve birth outcomes? Journal of Public Economics 72 135–154.
  • Ford, L. R. Jr. and Fulkerson, D. R. (1962). Flows in Networks. Princeton Univ. Press, Princeton, NJ.
    Mathematical Reviews (MathSciNet): MR159700
  • Foygel, R., Draisma, J. and Drton, M. (2012). Supplement to “Half-trek criterion for generic identifiability of linear structural equation models.” DOI:110.1214/12-AOS1012SUPP.
  • Garcia-Puente, L. D., Spielvogel, S. and Sullivant, S. (2010). Identifying causal effects with computer algebra. In Proceedings of the Twenty-sixth Conference on Uncertainty in Artificial Intelligence (UAI) (P. Grünwald and P. Spirtes, eds.). AUAI Press.
  • MathWorks Inc. (2010). MATLAB version 7.10.0 (R2010a). Natick, MA.
  • Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763–765.
    Mathematical Reviews (MathSciNet): MR331643
    Zentralblatt MATH: 0261.62043
    Digital Object Identifier: doi:10.1214/aos/1176342472
    Project Euclid: euclid.aos/1176342472
  • Pearl, J. (2000). Causality: Models, Reasoning, and Inference. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1744773
  • R Development Core Team. (2011). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria.
  • Richardson, T. and Spirtes, P. (2002). Ancestral graph Markov models. Ann. Statist. 30 962–1030.
    Mathematical Reviews (MathSciNet): MR1926166
    Zentralblatt MATH: 1033.60008
    Digital Object Identifier: doi:10.1214/aos/1031689015
    Project Euclid: euclid.aos/1031689015
  • Schrijver, A. (2004). Combinatorial Optimization. Polyhedra and Efficiency. Algorithms and Combinatorics 24 A. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1956924
  • Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press, Cambridge, MA.
    Mathematical Reviews (MathSciNet): MR1815675
  • Sullivant, S., Talaska, K. and Draisma, J. (2010). Trek separation for Gaussian graphical models. Ann. Statist. 38 1665–1685.
    Mathematical Reviews (MathSciNet): MR2662356
    Zentralblatt MATH: 1189.62091
    Digital Object Identifier: doi:10.1214/09-AOS760
    Project Euclid: euclid.aos/1269452651
  • Tian, J. (2005). Identifying direct causal effects in linear models. In Proceedings of the Twentieth National Conference on Artificial Intelligence (AAAI) 346–353. AAAI press, Palo Alto, CA.
  • Tian, J. (2009). Parameter identification in a class of linear structural equation models. In Proceedings of the Twenty-first International Joint Conference on Artificial Intelligence (IJCAI) 1970–1975. AAAI press, Palo Alto, CA.
  • Wermuth, N. (2011). Probability distributions with summary graph structure. Bernoulli 17 845–879.
    Mathematical Reviews (MathSciNet): MR2817608
    Digital Object Identifier: doi:10.3150/10-BEJ309
    Project Euclid: euclid.bj/1310042847
  • Wright, S. (1921). Correlation and causation. J. Agricultural Research 20 557–585.
  • Wright, S. (1934). The method of path coefficients. Ann. Math. Statist. 5 161–215.

Supplemental materials

  • Supplementary material: Inconclusive graphs, proofs and algorithms. The supplement starts with lists of some mixed graphs on $m=5$ nodes that are not classifiable using our methods, to illustrate the existing “gap” between our two criteria. After that we prove lemmas used in the main paper for establishing the HTC-identifiability and HTC-infinite-to-one criteria, and we provide details for the results relating HTC-identifiability to GC-identifiability and to graph decomposition. We then give correctness proofs for our algorithms for checking the HTC-criteria, and we discuss the weak HTC-criteria. The supplementary article concludes with a computational-algebraic discussion of the polynomial equations that led to the HTC-criteria.
    Digital Object Identifier: doi:10.1214/12-AOS1012SUPP
    Supplemental PDF (371 KB)

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