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Algebraic Aspects of Conditional Independence and Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Thomas Kahle, Johannes Rauh, Seth Sullivant
The goal of this chapter is to explore these questions and introduce tools from computational algebra for studying them. Our perspective is that, for a fixed type of random variable, the set of distributions that satisfy a collection of independence constraints is the zero set of a collection of polynomial equations. Solutions of systems of polynomial equations are the objects of study of algebraic geometry, and so tools from algebra can be brought to bear on the problem. The next section contains an overview of basic ideas in algebraic geometry which are useful for the study of conditional independence structures and graphical models. In particular, it introduces algebraic varieties, polynomial ideals, and primary decomposition. Section 3.3 introduces the ideals associated to families of conditional independence statements, and explains how to apply the basic techniques to deduce conditional independence implications. Section 3.4 illustrates the main ideas with some deeper examples coming from the literature. Section 3.5 concerns the vanishing ideal of a graphical model, which is a complete set of implicit restrictions for that model. This set of restrictions is usually much larger than the set of conditional independence constraints that come from the graph, but it can illuminate the structure of the model especially with more complex families of models involving mixed graphs or hidden random variables. Section 3.6 highlights some key references in this area.
On the Primary Decomposition of k-Ideals and Fuzzy k-Ideals in Semirings
Published in Fuzzy Information and Engineering, 2021
Ram Parkash Sharma, Madhu Dadhwal, Richa Sharma, S. Kar
Let I be a k-ideal of a semiring R. Then I is said to have a primary decomposition if I can be expressed as , where each is a primary ideal of R. Also, a primary decomposition of the type with , is called a reduced primary decomposition of I, if 's are distinct and I cannot be expressed as an intersection of a proper subset of ideals in the primary decomposition of I. A reduced primary decomposition can be obtained from any primary decomposition by deleting those that contains and grouping together all distinct 's-primary ideal.