The Annals of Statistics

The maximum likelihood threshold of a path diagram

Mathias Drton, Christopher Fox, Andreas Käufl, and Guillaume Pouliot

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Abstract

Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.

Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1536-1553.

Dates
Received: February 2018
First available in Project Euclid: 13 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1550026848

Digital Object Identifier
doi:10.1214/18-AOS1724

Mathematical Reviews number (MathSciNet)
MR3911121

Zentralblatt MATH identifier
07053517

Subjects
Primary: 62H12: Estimation 62J05: Linear regression

Keywords
Covariance matrix graphical model maximum likelihood normal distribution path diagram structural equation model

Citation

Drton, Mathias; Fox, Christopher; Käufl, Andreas; Pouliot, Guillaume. The maximum likelihood threshold of a path diagram. Ann. Statist. 47 (2019), no. 3, 1536--1553. doi:10.1214/18-AOS1724. https://projecteuclid.org/euclid.aos/1550026848


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References

  • Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
  • Andersson, S. A., Madigan, D. and Perlman, M. D. (1997). A characterization of Markov equivalence classes for acyclic digraphs. Ann. Statist. 25 505–541.
  • Barber, R. F., Drton, M. and Weihs, L. (2015). SEMID: Identifiability of linear structural equation models. R package version 0.2.
  • Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, Chichester.
  • Buhl, S. L. (1993). On the existence of maximum likelihood estimators for graphical Gaussian models. Scand. J. Stat. 20 263–270.
  • Dahl, J., Vandenberghe, L. and Roychowdhury, V. (2008). Covariance selection for nonchordal graphs via chordal embedding. Optim. Methods Softw. 23 501–520.
  • Diestel, R. (2010). Graph Theory, 4th ed. Graduate Texts in Mathematics 173. Springer, Heidelberg.
  • Drton, M. (2009). Likelihood ratio tests and singularities. Ann. Statist. 37 979–1012.
  • Drton, M. (2016). Algebraic problems in structural equation modeling. Available at arXiv:1612.05994.
  • Drton, M., Fox, C. and Wang, Y. S. (2018). Computation of maximum likelihood estimates in cyclic structural equation models. Ann. Statist. To appear. Available at arXiv:1610.03434.
  • Drton, M. and Richardson, T. S. (2008). Graphical methods for efficient likelihood inference in Gaussian covariance models. J. Mach. Learn. Res. 9 893–914.
  • Drton, M. and Yu, J. (2010). On a parametrization of positive semidefinite matrices with zeros. SIAM J. Matrix Anal. Appl. 31 2665–2680.
  • Evans, R. J. and Richardson, T. S. (2016). Smooth, identifiable supermodels of discrete DAG models with latent variables. Available at arXiv:1511.06813.
  • Foygel, R., Draisma, J. and Drton, M. (2012). Half-trek criterion for generic identifiability of linear structural equation models. Ann. Statist. 40 1682–1713.
  • Grace, J. B., Anderson, T. M., Seabloom, E. W., Borer, E. T., Adler, P. B., Harpole, W. S., Hautier, Y., Hillebrand, H., Lind, E. M., Pärtel, M., Bakker, J. D., Buckley, Y. M., Crawley, M. J., Damschen, E. I., Davies, K. F., Fay, P. A., Firn, J., Gruner, D. S., Hector, A., Knops, J. M. H., MacDougall, A. S., Melbourne, B. A., Morgan, J. W., Orrock, J. L., Prober, S. M. and Smith, M. D. (2016). Integrative modelling reveals mechanisms linking productivity and plant species richness. Nature 529 390–393.
  • Grone, R., Johnson, C. R., de Sá, E. M. and Wolkowicz, H. (1984). Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58 109–124.
  • Gross, E. and Sullivant, S. (2018). The maximum likelihood threshold of a graph. Bernoulli 24 386–407.
  • Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge.
  • Kauermann, G. (1996). On a dualization of graphical Gaussian models. Scand. J. Stat. 23 105–116.
  • Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Clarendon Press, New York.
  • Maathuis, M. H., Colombo, D., Kalisch, M. and Bühlmann, P. (2010). Predicting causal effects in large-scale systems from observational data. Nat. Methods 7 247–248.
  • Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form in a multivariate sample. Ann. Statist. 1 763–765.
  • Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge Univ. Press, Cambridge.
  • Spirtes, P., Glymour, C. and Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press, Cambridge, MA.
  • Uhler, C. (2012). Geometry of maximum likelihood estimation in Gaussian graphical models. Ann. Statist. 40 238–261.
  • Woodbury, M. A. (1950). Inverting Modified Matrices. Statistical Research Group, Memo. Rep. 42. Princeton Univ., Princeton, NJ.
  • Wright, S. (1921). Correlation and causation. J. Agricultural Research 20 557–585.
  • Wright, S. (1934). The method of path coefficients. Ann. Math. Stat. 5 161–215.