Approximations in the homogeneous Ising model

TL;DR

This paper introduces scalable approximation formulas for key intractable quantities in the homogeneous Ising model, enabling practical Bayesian inference and hypothesis testing in large-scale applications.

Contribution

It provides novel, accurate approximation methods for intractable quantities in the homogeneous Ising model, facilitating inference in large and complex systems.

Findings

Approximation formulas perform well in simulations.

Methods are scalable to large graphs.

Applications include fMRI analysis and agricultural yield studies.

Abstract

The Ising model is important in statistical modeling and inference in many applications, however its normalizing constant, mean number of active vertices and mean spin interaction -- quantities needed in inference -- are computationally intractable. We provide accurate approximations that make it possible to numerically calculate these quantities in the homogeneous case. Simulation studies indicate good performance of our approximation formulae that are scalable and unfazed by the size (number of nodes, degree of graph) of the Markov Random Field. The practical import of our approximation formulae is illustrated in performing Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment, and also in likelihood ratio testing for anisotropy in the spatial patterns of yearly increases in pistachio tree yields.