only one single mode, “the paramagnetic state” is selected. The Hopfield model, introduced originally to model auto-associative memories, is a special case of an Ising model, where the coupling matrix is of low rank $p\le N$ and corresponds to the sum of outer products of $p$ given spin vectors $\{\xi^k,k=1\ldots p\}$, each one representing a specific attractive pattern: $$J_{ij} = \frac{1}{p}\sum_{k=1}^N\xi_i^k\xi_j^k$$ In our inference context, these patterns are not given directly, the input of the model being the covariance matrix. In [@CoMoSe] these couplings are interpreted as the contribution stemming from the $p$ largest principle axes of the correlation matrix. This lead in particular the authors to propose an extension of the Hopfield model by introducing repulsive patterns to take as well into account the smallest principal axes. Assuming small patterns coefficients $|\xi^k| <1/\sqrt{N}$, they come up with the following couplings with highest likelihood: $$J_{ij}^{Hopfield} \equiv \frac{1}{\sqrt{(1-m_i^2)(1-m_j^2)}}
\sum_{k=1}^p\Bigl(\bigl(1-\frac{1}{\lambda_k}\bigr)v_i^kv_j^k - \bigl(\frac{1}{\lambda_{N-k}}-1\bigr)v_i^{N-k}v_j^{N-k}\Bigr)$$ at first order of the perturbation. At this order of approximation the local fields are given by $$h_i = \tanh(m_i)-\sum_j J_{ij}^{Hopfield}m_j.$$ In a previous study [@FuLaAu] we found a connection between the plain direct BP method with the Hopfield model, by considering a $1$-parameter deformation of the Bethe inference model 