The Ising distribution as a latent variable model

We show that the Ising distribution can be treated as a latent variable model, where a set of N real-valued, correlated random variables are drawn and used to generate N binary spins independently. This allows to approximate the Ising distribution by a simpler model where the latent variables follow a multivariate normal distribution. The resulting approximation bears similarities with the Thouless Anderson Palmer (TAP) solution from mean field theory, but retains a broader range of applicability when the coupling weights are not independently distributed. Moreover, unlike classic mean field approaches, the approximation can be used to generate correlated spin patterns.