Annals of Statistics

Geometry of maximum likelihood estimation in Gaussian graphical models

Caroline Uhler

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We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.

Article information

Ann. Statist., Volume 40, Number 1 (2012), 238-261.

First available in Project Euclid: 29 March 2012

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Zentralblatt MATH identifier

Primary: 62H12: Estimation 14Q10: Surfaces, hypersurfaces 90C25: Convex programming

Gaussian graphical model maximum likelihood estimation matrix completion problems duality algebraic statistics algebraic variety number of observations sufficient statistics treewidth elimination ideal ML degree bipartite graphs


Uhler, Caroline. Geometry of maximum likelihood estimation in Gaussian graphical models. Ann. Statist. 40 (2012), no. 1, 238--261. doi:10.1214/11-AOS957.

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  • [1] Acquistapace, F., Broglia, F. and Vélez, M. P. (1999). Basicness of semialgebraic sets. Geom. Dedicata 78 229–240.
  • [2] Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. Wiley, Chichester.
  • [3] Barrett, W., Johnson, C. R. and Tarazaga, P. (1993). The real positive definite completion problem for a simple cycle. Linear Algebra Appl. 192 3–31.
  • [4] Barrett, W. W., Johnson, C. R. and Loewy, R. (1996). The real positive definite completion problem: Cycle completability. Mem. Amer. Math. Soc. 122 viii+69.
  • [5] Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series 9. IMS, Hayward, CA.
  • [6] Buhl, S. L. (1993). On the existence of maximum likelihood estimators for graphical Gaussian models. Scand. J. Stat. 20 263–270.
  • [7] Cox, D., Little, J. and O’Shea, D. (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, New York.
  • [8] Dempster, A. P. (1972). Covariance selection. Biometrics 28 157–175.
  • [9] Drton, M., Sturmfels, B. and Sullivant, S. (2009). Lectures on Algebraic Statistics. Oberwolfach Seminars 39. Birkhäuser, Basel.
  • [10] Frets, G. P. (1921). Heredity of head form in man. Genetica 3 193–400.
  • [11] Gehrmann, H. and Lauritzen, S. L. (2011). Estimation of means in graphical Gaussian models with symmetries. Preprint. Available at
  • [12] Grant, M. and Boyd, S. CVX, a Matlab software for disciplined convex programming. Available at
  • [13] Grayson, D. R. and Stillman, M. E. Macaulay2, a software system for research in algebraic geometry. Available at
  • [14] 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.
  • [15] Hastie, T., Tibshirani, R. and Friedman, J. (2009). The Elements of Statistical Learning, 2nd ed. Springer Series in Statistics. Springer, New York.
  • [16] Højsgaard, S. and Lauritzen, S. L. (2008). Graphical Gaussian models with edge and vertex symmetries. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 1005–1027.
  • [17] Lauritzen, S. L. (1996). Graphical Models. Clarendon, Oxford.
  • [18] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • [19] Schäfer, J. and Strimmer, K. (2005). Learning large-scale graphical Gaussian models from genomic data. In Science of Complex Networks: From Biology to the Internet and WWW. The American Institute of Physics, College Park, MD.
  • [20] Sturmfels, B. and Uhler, C. (2010). Multivariate Gaussian, semidefinite matrix completion, and convex algebraic geometry. Ann. Inst. Statist. Math. 62 603–638.
  • [21] Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, Chichester.
  • [22] Wu, X., Ye, Y. and Subramanian, K. R. (2003). Interactive analysis of gene interactions using graphical Gaussian model. ACM SIGKDD Workshop on Data Mining in Bioinformatics 3 63–69.