  State-space models (SSM) are central to describe time-varying complex systems
in countless signal processing applications such as remote sensing, networks,
biomedicine, and finance to name a few. Inference and prediction in SSMs are
possible when the model parameters are known, which is rarely the case. The
estimation of these parameters is crucial, not only for performing statistical
analysis, but also for uncovering the underlying structure of complex
phenomena. In this paper, we focus on the linear-Gaussian model, arguably the
most celebrated SSM, and particularly in the challenging task of estimating the
transition matrix that encodes the Markovian dependencies in the evolution of
the multi-variate state. We introduce a novel perspective by relating this
matrix to the adjacency matrix of a directed graph, also interpreted as the
causal relationship among state dimensions in the Granger-causality sense.
Under this perspective, we propose a new method called GraphEM based on the
well sounded expectation-maximization (EM) methodology for inferring the
transition matrix jointly with the smoothing/filtering of the observed data. We
propose an advanced convex optimization solver relying on a consensus-based
implementation of a proximal splitting strategy for solving the M-step. This
approach enables an efficient and versatile processing of various sophisticated
priors on the graph structure, such as parsimony constraints, while benefiting
from convergence guarantees. We demonstrate the good performance and the
interpretable results of GraphEM by means of two sets of numerical examples.
