Algebraic geometry of Gaussian Bayesian networks

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.Comment: 30 page, 4 figure

Combinatorial symbolic powers

Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gr\"obner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gr\"obner bases of symbolic powers of determinantal and Pfaffian ideals.Comment: 29 pages, 3 figures, Positive characteristic results incorporated into main body of pape

Toric fiber products

We introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading. Under the assumption that the set of degrees of the variables form a linearly independent set, we can explicitly describe generating sets and Groebner bases for these ideals. This allows us to unify and generalize some results in algebraic statistics.Comment: 19 page

The Maximum Likelihood Threshold of a Graph

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $n-1$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold.Comment: Added Section 6 and Section