Before revision:  The approximation is formally related to an advanced mean field technique known as adaptive TAP, and its domain of validity is similar. When valid, it allows a principled replacement of the Ising distribution by a distribution much easier to sample and manipulate .
After revision:  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' .