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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
Non-vanishing of Betti numbers of edge ideals and complete bipartite subgraphs
Given a finite simple graph one can associate the edge ideal. In this paper
we prove that a graded Betti number of the edge ideal does not vanish if the
original graph contains a set of complete bipartite subgraphs with some
conditions. Also we give a combinatorial description for the projective
dimension of the edge ideals of unmixed bipartite graphs.Comment: 19 pages; v2: we added Section 7 and revised mainly Sections 5 and 6;
v3 improves the exposition throughou
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