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Analyzing multi-layered graphical models provides insight into understandingthe conditional relationships among nodes within layers after adjusting for andquantifying the effects of nodes from other layers. We obtain the penalizedmaximum likelihood estimator for Gaussian multi-layered graphical models, basedon a computational approach involving screening of variables, iterativeestimation of the directed edges between layers and undirected edges withinlayers and a final refitting and stability selection step that providesimproved performance in finite sample settings. We establish the consistency ofthe estimator in a high-dimensional setting. To obtain this result, we developa strategy that leverages the biconvexity of the likelihood function to ensureconvergence of the developed iterative algorithm to a stationary point, as wellas careful uniform error control of the estimates over iterations. Theperformance of the maximum likelihood estimator is illustrated on syntheticdata.

Penalized Maximum Likelihood Estimation of Multi-layered Gaussian Graphical Models