  In this paper, we introduce a new directed graphical model from Gaussian
data: the Gaussian graphical interaction model (GGIM). The development of this
model 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 model
and provide interpretations of the edges in the 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.
Furthermore, our approach leads to a natural definition of directed conditional
independence of two elements in a stationary Gaussian process. We demonstrate
that the problem of learning sparse GGIMs 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. Finally, we consider the
problem of learning GGIMs associated with sparse directed conditional
dependence relationships. In all, the new model presents a novel perspective on
directed relationships between variables and significantly expands on the state
of the art in Gaussian graphical modeling.
