Ising distribution as a latent variable model

Adrien Wohrer (1)

1 IP - Institut Pascal

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 un-practical, 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'.

Domaines: Mécanique statistique [cond-mat.stat-mech]; Systèmes désordonnés et réseaux de neurones [cond-mat.dis-nn]; Neurosciences

Adrien Wohrer. Ising distribution as a latent variable model. Physical Review E , 2019, 99 (4), pp.042147. ⟨10.1103/PhysRevE.99.042147⟩. ⟨hal-02170202⟩