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arXiv · 2510.12570

Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach

Emiliano Liwski
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Abstract

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.

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Emiliano Liwski. 2025-10-14. Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach. https://arxiv.org/abs/2510.12570

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