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Lattice supported distributions and graphical models

Thomas Kahle, Seth Sullivant

Abstract

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices.

Lattice supported distributions and graphical models

Abstract

For the distributions of finitely many binary random variables, we study the interaction of restrictions of the supports with conditional independence constraints. We prove a generalization of the Hammersley-Clifford theorem for distributions whose support is a natural distributive lattice: that is, any distribution which has natural lattice support and satisfies the pairwise Markov statements of a graph must factor according to the graph. We also show a connection to the Hibi ideals of lattices.

Paper Structure

This paper contains 7 sections, 19 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on Distributive Lattices
  3. Ideals associated CI statements and graphical models
  4. Compatibility of Lattice Structure and Graph
  5. Limits of Distributions that Factorize
  6. The natural lattice supported Hammersley-Clifford theorem
  7. Vanishing Ideals of Lattice Supported Graphical Models

Key Result

Theorem 1.3

Let $G = ([m], E)$ be a graph and $p \in \mathbb{R}^{2^{[m]}}$ a probability distribution.

Figures (2)

  • Figure 1: A poset, its lattice of order ideals, and a graph appearing in Example \ref{['ex:master']}
  • Figure 2: An unnatural lattice support of a $4$-cycle quartic.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Lemma 2.6
  • ...and 40 more