Lattice supported distributions and graphical models

TL;DR

This paper generalizes the Hammersley-Clifford theorem to distributions supported on natural distributive lattices, linking support restrictions with conditional independence in graphical models.

Contribution

It extends the classical theorem to lattice-supported distributions and connects these to Hibi ideals, advancing the understanding of support constraints in graphical models.

Findings

Distributions with lattice support satisfy the pairwise Markov property and factor according to the graph.

Established a connection between lattice-supported distributions and Hibi ideals.

Generalized the Hammersley-Clifford theorem for a broader class of distributions.

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.