Name
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. |
Year | DOI | Venue |
|---|---|---|
2008 | 10.1016/j.aam.2007.04.004 | Advances in Applied Mathematics |
Keywords | Field | DocType |
gaussian case,conditional independence statement,bayesian network,classical construction,conditional independence model,hidden variable model,hidden variable,algebraic variety,underlying graph,algebraic geometry,gaussian bayesian network,multivariate gaussian,algebraic statistics,graphical model,conditional independence,hidden variables,positive definite,schubert variety,grobner basis,random variable | Combinatorics,Algebraic geometry,Algebra,Conditional independence,Mathematical analysis,Bayesian network,Algebraic variety,Gröbner basis,Graphical model,Schubert variety,Algebraic statistics,Mathematics | Journal |
Volume | Issue | ISSN |
40 | 4 | 0196-8858 |
Citations | PageRank | References |
5 | 0.69 | 3 |
Authors | ||
1 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Seth Sullivant | 1 | 93 | 19.17 |