Cyclic directed graphs are rather ubiquitous in modeling economic processes and natural systems, e.g., in the field of biology. In fact, feedback loops in causal systems are generally helpful to improve system performance in the presence of model uncertainty and achieve stability of the system. However, there are relatively few works on characterizing and learning structures that contain cycles. In many state-of-the-art causal models, not only is feedback ignored, but it is also explicitly assumed that there are no cycles passing information among the considered quantities. This discrimination against cyclic structures in the literature is primarily due to simplicities of working with acyclic models and the fact that even a generally accepted definition of statistical equivalence does not exist in the literature for cyclic directed graphs.

The main way for defining equivalence among acyclic directed graphs is based on the conditional independencies of the distributions that they can generate. However, it is known that when cycles are allowed in the structure, conditional independence is not a suitable notion for equivalence of two structures, as it does not reflect all the information in the distribution that can be used for identification of the underlying structure. In this project, we present a general, unified notion of equivalence for linear Gaussian directed graphs.

The main way for defining equivalence among acyclic directed graphs is based on the conditional independencies of the distributions that they can generate. However, it is known that when cycles are allowed in the structure, conditional independence is not a suitable notion for equivalence of two structures, as it does not reflect all the information in the distribution that can be used for identification of the underlying structure. In this project, we present a general, unified notion of equivalence for linear Gaussian directed graphs.

**Publications:****A. Ghassami**, A.Yang, N. Kiyavash, and K. Zhang, “Characterizing Distribution Equivalence for Cyclic and Acyclic Directed Graphs,” under submission. arXiv preprint arXiv:1910.12993.