  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'.
