Directed Gaussian graphical models with toric vanishing ideals

TL;DR

This paper characterizes directed Gaussian graphical models with toric vanishing ideals, providing combinatorial criteria and analyzing their algebraic properties using the shortest trek map.

Contribution

It introduces criteria for constructing DAGs with toric vanishing ideals and studies their generating sets, advancing understanding of their algebraic structure.

Findings

Identifies combinatorial conditions for toric vanishing ideals in DAGs

Provides results on the generators of these toric ideals

Highlights the role of the shortest trek map in model characterization

Abstract

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.