Constraint-Based Causal Discovery In The Presence Of Cycles

While feedback loops are known to play important roles in many complex systems (for example, in economical, biological, chemical, physical, control and climatological systems), their existence is ignored in most of the causal discovery literature, where systems are typically assumed to be acyclic from the outset. When applying causal discovery algorithms designed for the acyclic setting on data generated by a system that involves feedback, one would not expect to obtain correct results, even in the infinite-sample limit. In this work, we show that---surprisingly---the output of the Fast Causal Inference (FCI) algorithm is correct if it is applied to observational data generated by a system that involves feedback. More specifically, we prove that for observational data generated by a simple and $\sigma$-faithful Structural Causal Model (SCM), FCI can be used to consistently estimate (i) the presence and absence of causal relations, (ii) the presence and absence of direct causal relations, (iii) the absence of confounders, and (iv) the absence of specific cycles in the causal graph of the SCM.