75 research outputs found
Graphical models in Macaulay2
The Macaulay2 package GraphicalModels contains algorithms for the algebraic
study of graphical models associated to undirected, directed and mixed graphs,
and associated collections of conditional independence statements. Among the
algorithms implemented are procedures for computing the vanishing ideal of
graphical models, for generating conditional independence ideals of families of
independence statements associated to graphs, and for checking for identifiable
parameters in Gaussian mixed graph models. These procedures can be used to
study fundamental problems about graphical models.Comment: Several changes to address referee comments and suggestions. We will
eventually include this package in the standard distribution of Macaulay2.
But until then, the associated Macaulay2 file can be found at
http://www.shsu.edu/~ldg005/papers.htm
The Geometry of Statistical Models for Two-Way Contingency Tables with Fixed Odds Ratios
We study the geometric structure of the statistical models for two-by-two
contingency tables. One or two odds ratios are fixed and the corresponding
models are shown to be a portion of a ruled quadratic surface or a segment.
Some pointers to the general case of two-way contingency tables are also given
and an application to case-control studies is presented.Comment: References were not displaying properly in the previous versio
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
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