Structure Learning for Cyclic Linear Causal Models

Inferring the structure of a causal model with feedback loops from observational data is a notoriously difficult--if not impossible--problem, particularly if one also seeks to guard against presence of latent confounders [9, 29]. We consider this problem for linear causal models given by mixed graphs (or path diagrams) with directed and bidirected edges. As detailed in Section 2, the vertices of such a graph correspond to the observed variables, and the directed edges encode structural equations that relate these variables up to stochastic noise. The bidirected edges indicate possible correlations among the noise terms, as may be induced by latent confounders. Much work has gone into algorithms that exploit conditional independence relations for learning the structure of causal models, or rather suitable equivalence classes of graphs encoding this structure; see, e.g., [10, 17, 18, 24, 25] or also the review of Spirtes and Zhang in [21, §18]. While methods have been developed that use information about conditional independence relations also in settings with feedback loops or latent variables, there is an inherent limitation to this approach as causal models with feedback loops or latent variables can generally not be characterized using conditional independence constraints alone [8, 27, 31, 32]. Alternatively, structure learning can be approached using score-based search techniques; see, e.g., [3, 28, 30].