Algebraic geometry of Bayesian networks

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.

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...

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...

Introductory Geometry: Algebraic Geometry

Algebraic Geometry the Fundamental Ideas of Abstract Algebraic Geometry

ALGEBRAIC GEOMETRY 79 It may be advisable to give a special name to those varieties which admit every (allowable) ground field as field of definition. Obviously, these are the varieties which are defined over the prime field of the given characteristic p. I propose to call them universal varieties. The projective space and the Grassmannian varieties are examples of universal varieties. Another ...