Learning Directed Graphical Models from Gaussian Data
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In this paper, we introduce two new directed graphical models from Gaussian data: the Gaussian graphical interaction model (GGIM) and the Gaussian graphical conditional expectation model (GGCEM). The development of these models comes from considering stationary Gaussian processes on graphs, and leveraging the equations between the resulting steady-state covariance matrix and the Laplacian matrix representing the interaction graph. Through the presentation of conceptually straightforward theory, we develop the new models and provide interpretations of the edges in each graphical model in terms of statistical measures. We show that when restricted to undirected graphs, the Laplacian matrix representing a GGIM is equivalent to the standard inverse covariance matrix that encodes conditional dependence relationships. We demonstrate that the problem of learning sparse GGIMs and GGCEMs for a given observation set can be framed as a LASSO problem. By comparison with the problem of inverse covariance estimation, we prove a bound on the difference between the covariance matrix corresponding to a sparse GGIM and the covariance matrix corresponding to the l_1-norm penalized maximum log-likelihood estimate. In all, the new models present a novel perspective on directed relationships between variables and significantly expand on the state of the art in Gaussian graphical modeling.
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