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Penalized likelihood methods for estimation of sparse high-dimensional directed acyclic graphs

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  • Ali Shojaie
  • George Michailidis

Abstract

Directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other. Estimation of directed graphs from observational data is computationally NP-hard. In addition, directed graphs with the same structure may be indistinguishable based on observations alone. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs, when variables inherit a natural ordering. We study variable selection consistency of lasso and adaptive lasso penalties in high-dimensional sparse settings, and propose an error-based choice for selecting the tuning parameter. We show that although the lasso is only variable selection consistent under stringent conditions, the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions. Copyright 2010, Oxford University Press.

Suggested Citation

  • Ali Shojaie & George Michailidis, 2010. "Penalized likelihood methods for estimation of sparse high-dimensional directed acyclic graphs," Biometrika, Biometrika Trust, vol. 97(3), pages 519-538.
    • Handle: RePEc:oup:biomet:v:97:y:2010:i:3:p:519-538
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    File URL: http://hdl.handle.net/10.1093/biomet/asq038
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    Cited by:

    1. Cao, Xuan & Khare, Kshitij & Ghosh, Malay, 2020. "Consistent Bayesian sparsity selection for high-dimensional Gaussian DAG models with multiplicative and beta-mixture priors," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    2. Xiao Guo & Hai Zhang, 2020. "Sparse directed acyclic graphs incorporating the covariates," Statistical Papers, Springer, vol. 61(5), pages 2119-2148, October.
    3. Davide Altomare & Guido Consonni & Luca La Rocca, 2013. "Objective Bayesian Search of Gaussian Directed Acyclic Graphical Models for Ordered Variables with Non-Local Priors," Biometrics, The International Biometric Society, vol. 69(2), pages 478-487, June.
    4. Daniel Felix Ahelegbey & Monica Billio & Roberto Casarin, 2016. "Sparse Graphical Vector Autoregression: A Bayesian Approach," Annals of Economics and Statistics, GENES, issue 123-124, pages 333-361.
    5. Yang Ni & Francesco C. Stingo & Veerabhadran Baladandayuthapani, 2015. "Bayesian nonlinear model selection for gene regulatory networks," Biometrics, The International Biometric Society, vol. 71(3), pages 585-595, September.
    6. Xiao Guo & Hai Zhang & Yao Wang & Yong Liang, 2019. "Structure learning of sparse directed acyclic graphs incorporating the scale-free property," Computational Statistics, Springer, vol. 34(2), pages 713-742, June.
    7. Xu Wang & JinRong Wang & Michal Fečkan, 2020. "BP Neural Network Calculus in Economic Growth Modelling of the Group of Seven," Mathematics, MDPI, vol. 8(1), pages 1-11, January.
    8. Yan Zhou & Peter X.‐K. Song & Xiaoquan Wen, 2021. "Structural factor equation models for causal network construction via directed acyclic mixed graphs," Biometrics, The International Biometric Society, vol. 77(2), pages 573-586, June.
    9. Zhang, Hongmei & Huang, Xianzheng & Han, Shengtong & Rezwan, Faisal I. & Karmaus, Wilfried & Arshad, Hasan & Holloway, John W., 2021. "Gaussian Bayesian network comparisons with graph ordering unknown," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    10. Davide Altomare & Guido Consonni & Luca La Rocca, 2011. "Objective Bayesian Search of Gaussian DAG Models with Non-local Priors," Quaderni di Dipartimento 140, University of Pavia, Department of Economics and Quantitative Methods.
    11. Abdullah Gov & Veli Yilanci, 2023. "Graphical Causality Test Approach to the Relationship Between Economic Growth, Energy Consumption, Foreign Trade Balance and Financial Development," Istanbul Journal of Economics-Istanbul Iktisat Dergisi, Istanbul University, Faculty of Economics, vol. 73(73-1), pages 203-230, June.
    12. Choi, Semin & Kim, Yesool & Park, Gunwoong, 2023. "Densely connected sub-Gaussian linear structural equation model learning via ℓ1- and ℓ2-regularized regressions," Computational Statistics & Data Analysis, Elsevier, vol. 181(C).
    13. Lee, Kyoungjae & Cao, Xuan, 2022. "Bayesian joint inference for multiple directed acyclic graphs," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    14. Shojaie Ali & Michailidis George, 2010. "Network Enrichment Analysis in Complex Experiments," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 9(1), pages 1-36, May.
    15. Bingkai Wang & Brian S. Caffo & Xi Luo & Chin‐Fu Liu & Andreia V. Faria & Michael I. Miller & Yi Zhao & for the Alzheimer's Disease Neuroimaging Initiative*, 2022. "Regularized regression on compositional trees with application to MRI analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(3), pages 541-561, June.
    16. Kuang‐Yao Lee & Lexin Li, 2022. "Functional structural equation model," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 600-629, April.
    17. Guido Consonni & Luca La Rocca, 2010. "Moment Priors for Bayesian Model Choice with Applications to Directed Acyclic Graphs," Quaderni di Dipartimento 115, University of Pavia, Department of Economics and Quantitative Methods.
    18. Baek, Changryong & Gates, Katheleen M. & Leinwand, Benjamin & Pipiras, Vladas, 2021. "Two sample tests for high-dimensional autocovariances," Computational Statistics & Data Analysis, Elsevier, vol. 153(C).

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