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TitleAlgebraic geometry of Gaussian Bayesian networks
Author(s) Seth Sullivant
TypeArticle in Journal
AbstractConditional independence models in the Gaussian case are algebraic varieties in the cone of positive definite covariance matrices. We study these varieties in the case of Bayesian networks, with a view towards generalizing the recursive factorization theorem to situations with hidden variables. In the case when the underlying graph is a tree, we show that the vanishing ideal of the model is generated by the conditional independence statements implied by graph. We also show that the ideal of any Bayesian network is homogeneous with respect to a multigrading induced by a collection of upstream random variables. This has a number of important consequences for hidden variable models. Finally, we relate the ideals of Bayesian networks to a number of classical constructions in algebraic geometry including toric degenerations of the Grassmannian, matrix Schubert varieties, and secant varieties.
KeywordsBayesian network, Graphical model, Algebraic statistics, Multivariate Gaussian, Gröbner basis
URL http://www.sciencedirect.com/science/article/pii/S0196885807000760
JournalAdvances in Applied Mathematics
Pages482 - 513
Translation No
Refereed No