  We consider the problem of structure learning for linear causal models based
on observational data. We treat models given by possibly cyclic mixed graphs,
which allow for feedback loops and effects of latent confounders. Generalizing
related work on bow-free acyclic graphs, we assume that the underlying graph is
simple. This entails that any two observed variables can be related through at
most one direct causal effect and that (confounding-induced) correlation
between error terms in structural equations occurs only in absence of direct
causal effects. We show that, despite new subtleties in the cyclic case, the
considered simple cyclic models are of expected dimension and that a previously
considered criterion for distributional equivalence of bow-free acyclic graphs
has an analogue in the cyclic case. Our result on model dimension justifies in
particular score-based methods for structure learning of linear Gaussian mixed
graph models, which we implement via greedy search.
