Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Emiliano Liwski

TL;DR

The paper develops a matroid-theoretic framework for decomposing CI ideals with hidden variables by linking determinantal CI conditions to circuit and matroid varieties. Introducing quasi-paving matroids, it proves that irreducible components of relevant varieties are realizable as matroid varieties $V_M$, and it provides a decomposition along nice partitions of dependent hyperplanes. Specializations to grid-like matroids $G_{k,l}$ and line-configuration matroids $L_n$ yield explicit decompositions and, for certain parameters, complete descriptions, together with generating functions that count irreducible components. The work integrates algebraic statistics with combinatorial geometry, delivering concrete counting tools and structural insights for CI models and their algebraic invariants.

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.

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

TL;DR

The paper develops a matroid-theoretic framework for decomposing CI ideals with hidden variables by linking determinantal CI conditions to circuit and matroid varieties. Introducing quasi-paving matroids, it proves that irreducible components of relevant varieties are realizable as matroid varieties , and it provides a decomposition along nice partitions of dependent hyperplanes. Specializations to grid-like matroids and line-configuration matroids yield explicit decompositions and, for certain parameters, complete descriptions, together with generating functions that count irreducible components. The work integrates algebraic statistics with combinatorial geometry, delivering concrete counting tools and structural insights for CI models and their algebraic invariants.

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.
Paper Structure (15 sections, 33 theorems, 84 equations, 6 figures, 2 tables)

This paper contains 15 sections, 33 theorems, 84 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Preliminaries
  4. Matroids
  5. Paving matroids
  6. Nilpotent and liftable paving matroids
  7. Quasi-paving matroids
  8. Decomposition of circuit varieties
  9. Decomposition of CI Ideals
  10. Variety from n lines in P2
  11. Number of irreducible components
  12. Generating function for the number of vector partitions
  13. Counting the irreducible components of $V_{\Delta}$
  14. Counting the irreducible components of $V_{\mathcal{C}(L_{n})}$
  15. Proof of Theorem \ref{['Theorem']}

Key Result

Theorem 1.4

clarke2021matroid Let $s, t, k, l, n$ be positive integers satisfying $3 \le s \le t \le l$, $s \le k$, and $t \le n \le s+t-3$. Then defines the collection of circuits of a realizable matroid $M$ on $[kl]$ of rank $n$, where $\min$ denotes the inclusion-minimal subsets. Moreover, the corresponding varieties coincide:

Figures (6)

  • Figure 1: (Left) Uniform matroid $U_{2,7}$; (Right) Fano plane.
  • Figure 2: (Left) Three concurrent lines; (Right) Quadrilateral set.
  • Figure 3: Two $3$-quasi-paving matroids
  • Figure 4: (Left) $3\times 3$ grid; (Right) $3\times 4$ grid
  • Figure 5: (Left) Matroid $G_{3,5}$; (Right) Matroid $G_{4,5}$.
  • ...and 1 more figures

Theorems & Definitions (98)

  • Definition 1.1: CI ideal
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem (A)
  • Theorem (B)
  • Theorem (C)
  • Theorem (D)
  • ...and 88 more