Lattice Conditional Independence Models and Hibi Ideals

TL;DR

This paper links lattice-based conditional independence models with Hibi ideals, providing a new algebraic perspective that generalizes Gaussian models and connects to causal graphs, with applications in statistics and information theory.

Contribution

It establishes a novel correspondence between Lattice Conditional Independence models and Hibi ideals, extending the theory beyond Gaussian cases using algebraic structures.

Findings

Lattice CI models can be described via Hibi ideals.

Directed acyclic graphs correspond to chains in the lattice.

Hibi ideals can be recovered from bipartite graphs via Alexander duality.

Abstract

Lattice Conditional Independence models are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the LCI models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log-linear models, time series, and Shannon information flow.