Algebraic geometry of Gaussian Bayesian networks

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 gen...

Algebraic geometry of Bayesian networks

We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties.

An Algebraic Characterization of Equivalent Bayesian Networks

In this paper, we propose an algebraic characterization for equivalent classes of Bayesian networks. Unlike the other characterizations, which are based on the graphical structure of Bayesian networks, our algebraic characterization is derived from the intrinsic algebraic structure of Bayesian networks, i.e., joint probability distribution factorization. The new proposed algebraic characterizat...

Local Propagation in Conditional Gaussian Bayesian Networks

This paper describes a scheme for local computation in conditional Gaussian Bayesian networks that combines the approach of Lauritzen and Jensen (2001) with some elements of Shachter and Kenley (1989). Message passing takes place on an elimination tree structure rather than the more compact (and usual) junction tree of cliques. This yields a local computation scheme in which all calculations in...

Introductory Geometry: Algebraic Geometry