2,202 research outputs found
Binomial edge ideals and conditional independence statements
AbstractWe introduce binomial edge ideals attached to a simple graph G and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gröbner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gröbner basis for general G. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of G. We provide sufficient conditions for Cohen–Macaulayness for closed and nonclosed graphs.Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation
Algebraic Aspects of Conditional Independence and Graphical Models
This chapter of the forthcoming Handbook of Graphical Models contains an
overview of basic theorems and techniques from algebraic geometry and how they
can be applied to the study of conditional independence and graphical models.
It also introduces binomial ideals and some ideas from real algebraic geometry.
When random variables are discrete or Gaussian, tools from computational
algebraic geometry can be used to understand implications between conditional
independence statements. This is accomplished by computing primary
decompositions of conditional independence ideals. As examples the chapter
presents in detail the graphical model of a four cycle and the intersection
axiom, a certain implication of conditional independence statements. Another
important problem in the area is to determine all constraints on a graphical
model, for example, equations determined by trek separation. The full set of
equality constraints can be determined by computing the model's vanishing
ideal. The chapter illustrates these techniques and ideas with examples from
the literature and provides references for further reading.Comment: 20 pages, 1 figur
Robustness and Conditional Independence Ideals
We study notions of robustness of Markov kernels and probability distribution
of a system that is described by input random variables and one output
random variable. Markov kernels can be expanded in a series of potentials that
allow to describe the system's behaviour after knockouts. Robustness imposes
structural constraints on these potentials. Robustness of probability
distributions is defined via conditional independence statements. These
statements can be studied algebraically. The corresponding conditional
independence ideals are related to binary edge ideals. The set of robust
probability distributions lies on an algebraic variety. We compute a Gr\"obner
basis of this ideal and study the irreducible decomposition of the variety.
These algebraic results allow to parametrize the set of all robust probability
distributions.Comment: 16 page
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
Generalized Binomial Edge Ideals
This paper studies a class of binomial ideals associated to graphs with
finite vertex sets. They generalize the binomial edge ideals, and they arise in
the study of conditional independence ideals. A Gr\"obner basis can be computed
by studying paths in the graph. Since these Gr\"obner bases are square-free,
generalized binomial edge ideals are radical. To find the primary decomposition
a combinatorial problem involving the connected components of subgraphs has to
be solved. The irreducible components of the solution variety are all rational.Comment: 6 pages. arXiv admin note: substantial text overlap with
arXiv:1110.133
- …