# Ising distribution as a latent variable model

## TL;DR

This paper demonstrates that the Ising distribution can be viewed as a latent variable model and can be approximated by a Cox distribution under certain conditions, linking it to mean-field methods.

## Contribution

It shows how the Ising distribution can be replaced by a Cox distribution using a latent variable perspective, especially within the mean-field domain.

## Key findings

- Ising distribution can be treated as a latent variable model.
- A variational approach links the Ising model to mean-field methods.
- Numerical tests confirm the conditions for replacement in practical applications.

## Abstract

During the past decades, the Ising distribution has attracted interest in many applied disciplines, as the maximum entropy distribution associated to any set of correlated binary (`spin') variables with observed means and covariances. However, numerically speaking, the Ising distribution is unpractical, so alternative models are often preferred to handle correlated binary data. One popular alternative, especially in life sciences, is the Cox distribution (or the closely related dichotomized Gaussian distribution and log-normal Cox point process), where the spins are generated independently conditioned on the drawing of a latent variable with a multivariate normal distribution. This article explores the conditions for a principled replacement of the Ising distribution by a Cox distribution. It shows that the Ising distribution itself can be treated as a latent variable model, and it explores when this latent variable has a quasi-normal distribution. A variational approach to this question reveals a formal link with classic mean-field methods, especially Opper and Winther's adaptive TAP approximation. This link is confirmed by weak coupling (Plefka) expansions of the different approximations and then by numerical tests. Overall, this study suggests that an Ising distribution can be replaced by a Cox distribution in practical applications, precisely when its parameters lie in the `mean-field domain'.