Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908592852959232 |
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| author | Liwski, Emiliano |
| author_facet | Liwski, Emiliano |
| contents | We study a class of determinantal ideals arising from conditional independence (CI) statements with hidden variables. Such CI statements translate into determinantal conditions on a matrix whose entries represent the probabilities of events involving the observed random variables. Our main objective is to determine the irreducible components of the corresponding varieties and to provide a combinatorial or geometric interpretation of each. We achieve this by introducing a new approach rooted in matroid theory. In particular, we introduce a new class of matroids, which we term quasi-paving matroids, and show that the components of these determinantal varieties are precisely matroid varieties of quasi-paving matroids. Moreover, we derive generating functions that encode the number of irreducible components of these CI ideals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach Liwski, Emiliano Combinatorics Commutative Algebra Algebraic Geometry We study a class of determinantal ideals arising from conditional independence (CI) statements with hidden variables. Such CI statements translate into determinantal conditions on a matrix whose entries represent the probabilities of events involving the observed random variables. Our main objective is to determine the irreducible components of the corresponding varieties and to provide a combinatorial or geometric interpretation of each. We achieve this by introducing a new approach rooted in matroid theory. In particular, we introduce a new class of matroids, which we term quasi-paving matroids, and show that the components of these determinantal varieties are precisely matroid varieties of quasi-paving matroids. Moreover, we derive generating functions that encode the number of irreducible components of these CI ideals. |
| title | Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach |
| topic | Combinatorics Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2510.12570 |