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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 models vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.
We propose two types of Quantile Graphical Models (QGMs) --- Conditional Independence Quantile Graphical Models (CIQGMs) and Prediction Quantile Graphical Models (PQGMs). CIQGMs characterize the conditional independence of distributions by evaluating
Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be descr
Measuring conditional independence is one of the important tasks in statistical inference and is fundamental in causal discovery, feature selection, dimensionality reduction, Bayesian network learning, and others. In this work, we explore the connect
We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear conditional mean
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with hidden random variables corresponds to a variety defined