Generating large Ising models with Markov structure via simple linear relations

TL;DR

This paper introduces hollow-tree structures for palindromic Ising models, enabling local assessment of dependencies and facilitating the generation of large models with Markov properties.

Contribution

It extends tree graph concepts to hollow trees for palindromic Ising models, linking independence, partial correlations, and model structure.

Findings

Hollow-tree structures accurately capture dependencies in palindromic Ising models.

Conditional correlations within prime graphs match partial correlations under certain conditions.

Application to longitudinal data demonstrates practical model fitting.

Abstract

We extend the notion of a tree graph to sequences of prime graphs which are cycles and edges and name these non-chordal graphs hollow trees. These structures are especially attractive for palindromic Ising models, which mimic a symmetry of joint Gaussian distributions. We show that for an Ising model all defining independences are captured by zero partial correlations and conditional correlations agree with partial correlations within each prime graph if and only if the model is palindromic and has a hollow-tree structure. This implies that the strength of dependences can be assessed locally. We use the results to find a well-fitting general Ising model with hollow-tree structure for a set of longitudinal data.