  Analyzing multi-layered graphical models provides insight into understanding
the conditional relationships among nodes within layers after adjusting for and
quantifying the effects of nodes from other layers. We obtain the penalized
maximum likelihood estimator for Gaussian multi-layered graphical models, based
on a computational approach involving screening of variables, iterative
estimation of the directed edges between layers and undirected edges within
layers and a final refitting and stability selection step that provides
improved performance in finite sample settings. We establish the consistency of
the estimator in a high-dimensional setting. To obtain this result, we develop
a strategy that leverages the biconvexity of the likelihood function to ensure
convergence of the developed iterative algorithm to a stationary point, as well
as careful uniform error control of the estimates over iterations. The
performance of the maximum likelihood estimator is illustrated on synthetic
data.
