  In this work we explain how to properly use mean-field methods to solve the
inverse Ising problem when the phase space is clustered, that is many states
are present. The clustering of the phase space can occur for many reasons, e.g.
when a system undergoes a phase transition. Mean-field methods for the inverse
Ising problem are typically used without taking into account the eventual
clustered structure of the input configurations and may led to very bad
inference (for instance in the low temperature phase of the Curie-Weiss model).
In the present work we explain how to modify mean-field approaches when the
phase space is clustered and we illustrate the effectiveness of the new method
on different clustered structures (low temperature phases of Curie-Weiss and
Hopfield models).
