2018/07/19 by Mathias Drton, Drton, Mathias, Elina Robeva +3
Computer Science · #Bayesian Modeling and Causal Inference #Data Management and Algorithms #FOS: Mathematics #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1807.07561
openalex publication_date 2018/07/19 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Directed graphical models specify noisy functional relationships among a\ncollection of random variables. In the Gaussian case, each such model\ncorresponds to a semi-algebraic set of positive definite covariance matrices.\nThe set is given via parametrization, and much work has gone into obtaining an\nimplicit description in terms of polynomial (in-)equalities. Implicit\ndescriptions shed light on problems such as parameter identification, model\nequivalence, and constraint-based statistical inference. For models given by\ndirected acyclic graphs, which represent settings where all relevant variables\nare observed, there is a complete theory: All conditional independence\nrelations can be found via graphical d-separation and are sufficient for an\nimplicit description. The situation is far more complicated, however, when some\nof the variables are hidden (or in other words, unobserved or latent). We\nconsider models associated to mixed graphs that capture the effects of hidden\nvariables through correlated error terms. The notion of trek separation\nexplains when the covariance matrix in such a model has submatrices of low rank\nand generalizes d-separation. However, in many cases, such as the infamous\nVerma graph, the polynomials defining the graphical model are not\ndeterminantal, and hence cannot be explained by d-separation or\ntrek-separation. In this paper, we show that these constraints often correspond\nto the vanishing of nested determinants and can be graphically explained by a\nnotion of restricted trek separation.\n