Graphical Inference in Non-Markovian Linear-Gaussian State-Space Models

State-space models (SSMs) are common tools in time-series analysis for inference and prediction. SSMs are versatile probabilistic models that allow for Bayesian inference by describing a (generally Markovian) latent process. However, the parameters of that latent process are often unknown and must be estimated. In this paper, we consider the parameter estimation in a SSM with a non-Markovian linear-Gaussian latent process. This process is described as a vector auto-regressive with p unknown matrices. Our algorithm LaGrangEM estimates these matrices through an expectation-maximization algorithm that exploits a graphical interpretation of the latent process in order to define prior knowledge about the unknown parameters. We connect the new algorithm with existing approaches such as Granger causality and graphical inference in SSMs. We discuss the strong potential of the algorithm to bring interpretability, e.g., in estimating causal relationships and their delays. The numerical experiments also show a superiority in performance.