Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. By assuming each observation to arise as a one-time cross-sectional snapshot of a temporal process in equilibrium, they allow to model dependence structures that may include feedback loops. Specifically, the graphical continuous Lyapunov model consists of steady-state distributions of multivariate Ornstein-Uhlenbeck processes where sparsity assumptions on the drift matrices are represented with a directed graph. These distributions are Gaussian with covariance matrices that are parametrized as solutions of the continuous Lyapunov equation. In this talk, I will motivate the Lyapunov models and present the conditional independence structure as well as results on identifiability of the drift parameters in specific cases.
