Constraints in Gaussian Graphical Models
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In this paper, we consider the problem of finding the constraints in bow-free acyclic directed mixed graphs (ADMGs). ADMGs are a generalisation of directed acyclic graphs (DAGs) that allow for certain latent variables. We first show that minimal generators for the ideal (G) containing all the constraints of a Gaussian ADMG G corresponds precisely to the pairs of non-adjacent vertices in G. The proof of this theorem naturally leads to an efficient algorithm that fits a bow-free Gaussian ADMG by maximum likelihood. In particular, we can test for the goodness of fit of a given data set to a bow-free ADMG.
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