Joint estimation of multiple graphical models. (English) Zbl 1214.62058

Summary: Gaussian graphical models explore dependence relationships between random variables, through the estimation of the corresponding inverse covariance matrices. We develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure. In this setting, estimating a single graphical model would mask the underlying heterogeneity, while estimating separate models for each category does not take advantage of the common structure. We propose a method that jointly estimates the graphical models corresponding to the different categories present in the data, aiming to preserve the common structure, while allowing for differences between the categories. This is achieved through a hierarchical penalty that targets the removal of common zeros in the inverse covariance matrices across categories. We establish the asymptotic consistency and sparsity of the proposed estimator in the high-dimensional case, and illustrate its performance on a number of simulated networks. An application to learning semantic connections between terms from webpages collected from computer science departments is included.


62H12 Estimation in multivariate analysis
05C90 Applications of graph theory
65C60 Computational problems in statistics (MSC2010)
62P99 Applications of statistics
62F12 Asymptotic properties of parametric estimators


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