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April 2013 $\ell_{0}$-penalized maximum likelihood for sparse directed acyclic graphs
Sara van de Geer, Peter Bühlmann
Ann. Statist. 41(2): 536-567 (April 2013). DOI: 10.1214/13-AOS1085


We consider the problem of regularized maximum likelihood estimation for the structure and parameters of a high-dimensional, sparse directed acyclic graphical (DAG) model with Gaussian distribution, or equivalently, of a Gaussian structural equation model. We show that the $\ell_{0}$-penalized maximum likelihood estimator of a DAG has about the same number of edges as the minimal-edge I-MAP (a DAG with minimal number of edges representing the distribution), and that it converges in Frobenius norm. We allow the number of nodes $p$ to be much larger than sample size $n$ but assume a sparsity condition and that any representation of the true DAG has at least a fixed proportion of its nonzero edge weights above the noise level. Our results do not rely on the faithfulness assumption nor on the restrictive strong faithfulness condition which are required for methods based on conditional independence testing such as the PC-algorithm.


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Sara van de Geer. Peter Bühlmann. "$\ell_{0}$-penalized maximum likelihood for sparse directed acyclic graphs." Ann. Statist. 41 (2) 536 - 567, April 2013.


Published: April 2013
First available in Project Euclid: 26 April 2013

zbMATH: 1267.62037
MathSciNet: MR3099113
Digital Object Identifier: 10.1214/13-AOS1085

Primary: 62F12
Secondary: 62F30

Keywords: Causal inference , faithfulness condition , Gaussian structural equation model , graphical modeling , high-dimensional inference

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 2 • April 2013
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