Positivity in Linear Gaussian Structural Equation Models

We study a notion of positivity of Gaussian directed acyclic graphical models corresponding to a non-negativity constraint on the coefficients of the associated structural equation model. We prove that this constraint is equivalent to the distribution being conditionally increasing in sequence (CIS)...

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Bibliographic Details
Published in:arXiv.org 2023-05
Main Authors: Lodhia, Asad, Jan-Christian Hütter, Uhler, Caroline, Zwiernik, Piotr
Format: Article
Language:English
Subjects:
Algorithms
Constraint modelling
Equivalence
Maximum likelihood estimation
Multivariate statistical analysis
Permutations
Random variables
Online Access:Get full text
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Summary:We study a notion of positivity of Gaussian directed acyclic graphical models corresponding to a non-negativity constraint on the coefficients of the associated structural equation model. We prove that this constraint is equivalent to the distribution being conditionally increasing in sequence (CIS), a well-known subclass of positively associated random variables. These distributions require knowledge of a permutation, a CIS ordering, of the nodes for which the constraint of non-negativity holds. We provide an algorithm and prove in the noise-less setting that a CIS ordering can be recovered when it exists. We extend this result to the noisy setting and provide assumptions for recovering the CIS orderings. In addition, we provide a characterization of Markov equivalence for CIS DAG models. Further, we show that when a CIS ordering is known, the corresponding class of Gaussians lies in a family of distributions in which maximum likelihood estimation is a convex problem.
ISSN:2331-8422