Details:
Title  Algebraic geometry of Gaussian Bayesian networks  Author(s)  Seth Sullivant  Type  Article in Journal  Abstract  Conditional 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.  Keywords  Bayesian network, Graphical model, Algebraic statistics, Multivariate Gaussian, Gröbner basis  ISSN  01968858 
URL 
http://www.sciencedirect.com/science/article/pii/S0196885807000760 
Language  English  Journal  Advances in Applied Mathematics  Volume  40  Number  4  Pages  482  513  Year  2008  Edition  0  Translation 
No  Refereed 
No 
