Comment on: Decomposition of structural learning about directed acyclic graphs [1]

Comment on: Decomposition of structural learning about directed acyclic graphs [1] Abstract We propose an alternative proof concerning necessary and sufficient conditions to split the problem of searching for d-separators and building the skeleton of a DAG into small problems for every node of a separation tree T. The proof is simpler than the original [1]. The same proof structure has been used in [2] for learning the structure of multivariate regression chain graphs (MVR CGs). Keywords: Conditional independence, Structural learning, Decomposition, Directed acyclic graph, d-separation tree 1. Introduction In this paper we consider directed acyclic graphs (DAGs) and largely use the terminology of [1], where the reader can also find further details. For the reader's convenience we just recall the definition of d-separation tree here: Definition 1.1. Corresponding author Email addresses: javidian@email.sc.edu (Mohammad Ali Javidian), mgv@cse.sc.edu (Marco Valtorta) Preprint submitted to ArXiv July 2, 2018 2. Necessary and sufficient condition for decomposing structural learning of DAGs First we give several lemmas (Lemma 2.1-2.4) from [1, Appendix A] to be used in the proof of the main theorem in [1, Theorem 1].