Intervention in undirected Ising graphs and the partition function

TL;DR

This paper explores intervention strategies in undirected Ising models, proposing a Curie-Weiss approximation for the partition function, which is computationally challenging, and demonstrating its accuracy through theoretical results and simulations.

Contribution

It introduces a Curie-Weiss based approximation for the partition function in undirected Ising graphs, especially effective when weights are uniform or sub-Gaussian.

Findings

Exact partition function when weights are equal within cliques

Exponential closeness of approximation for sub-Gaussian weights

Simulation results confirming theoretical accuracy

Abstract

Undirected graphical models have many applications in such areas as machine learning, image processing, and, recently, psychology. Psychopathology in particular has received a lot of attention, where symptoms of disorders are assumed to influence each other. One of the most relevant questions practically is on which symptom (node) to intervene to have the most impact. Interventions in undirected graphical models is equal to conditioning, and so we have available the machinery with the Ising model to determine the best strategy to intervene. In order to perform such calculations the partition function is required, which is computationally difficult. Here we use a Curie-Weiss approach to approximate the partition function in applications of interventions. We show that when the connection weights in the graph are equal within each clique then we obtain exactly the correct partition function. And if the weights vary according to a sub-Gaussian distribution, then the approximation is exponentially close to the correct one. We confirm these results with simulations.