Constraint-Based Causal Discovery using Partial Ancestral Graphs in the\n presence of Cycles

While feedback loops are known to play important roles in many complex\nsystems, their existence is ignored in a large part of the causal discovery\nliterature, as systems are typically assumed to be acyclic from the outset.\nWhen applying causal discovery algorithms designed for the acyclic setting on\ndata generated by a system that involves feedback, one would not expect to\nobtain correct results. In this work, we show that -- surprisingly -- the\noutput of the Fast Causal Inference (FCI) algorithm is correct if it is applied\nto observational data generated by a system that involves feedback. More\nspecifically, we prove that for observational data generated by a simple and\n$\\sigma$-faithful Structural Causal Model (SCM), FCI is sound and complete, and\ncan be used to consistently estimate (i) the presence and absence of causal\nrelations, (ii) the presence and absence of direct causal relations, (iii) the\nabsence of confounders, and (iv) the absence of specific cycles in the causal\ngraph of the SCM. We extend these results to constraint-based causal discovery\nalgorithms that exploit certain forms of background knowledge, including the\ncausally sufficient setting (e.g., the PC algorithm) and the Joint Causal\nInference setting (e.g., the FCI-JCI algorithm).\n