(\[eq:ansatz\]) $$\label{eq:bpalpha}
{\cal P}({{\mathbf{x}}}) = \prod_{i<j}\Bigl(\frac{{\hat p}_{ij}(x_i,x_j)}{{\hat p}(x_i){\hat p}(x_j)}\Bigr)^\alpha\prod_i{\hat p}_i(x_i),$$ with $\alpha\in[0,1]$. We observed indeed that when the data corresponds to some multi-modal measure with well separated components, this measure coincides asymptotically with an Hopfield model made only of attractive pattern, representative of each component of the underlying measure. $\alpha$ represents basically the inverse temperature of the model and is easy to calibrate in practice.

More on the Bethe susceptibility {#sec:cumulant}
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The explicit relation (\[eq:invchis\]) between susceptibility and inverse susceptibility coefficients is not the only one that can be obtained. In fact, it is the specific property of a singly connected factor graph that two variables $x_i$ and $x_j$, conditionally to a variable $x_k$ are independent if $k$ is on the path between $i$ and $j$ along the tree: $$p(x_i,x_j,x_k) = p(x_i\vert x_k)p(x_j\vert x_k)p(x_k) = \frac{p(x_i,x_k)p(x_j,x_k)}{p(x_k)}$$ Using the parametrization (\[eq:pi\],\[eq:pij\]) with $m_{ij}=m_im_j+\chi_{ij}$ yields immediately $$\label{eq:chirec}
\chi_{ij} = \frac{\chi_{ik}\chi_{jk}}{1-m_k^2},\qquad\forall\ k\in(i,j)\ \text{along}\ {\mathcal{T}}.$$ By recurrence we get, as noticed in e.g. [@Mora], given the path $i_0=i,i_1,\ldots,i_{n+1}=j$ between $i$ and $j$ along the tree ${\mathcal{T}}$ $$\chi_{ij} = \frac{\prod_{a=0}^n\chi_{i_ai_{a+1}}}{\prod_{a=1}^n(1-m_{i_a}^2)},$$ reflecting the factorization of the joint measure. This expression actually coincides with (\[eq:invchis\]) only on a tree. On a loopy graph, this last expression 