intuitive assertion concerning the optimal model with tree like factor graphs, valid for any type of MRF. Suppose that we are given a set of single and pairwise empirical marginals $\hat p_i$ and $\hat p_{ij}$ for a set of $N$ real variables $\{x_i,i=1\ldots N\}$. If we start from an empty graph with no link, the joint probability distribution is simply the product form $${\mathcal{P}}^{(0)}(x_i) = \prod_{i=1}^N \hat p_i(x_i).$$ Adding a link $(ij)$ to the empty graph is optimally done by multiplying ${\mathcal{P}}^{(0)}$ by $\hat p_{ij}/\hat p_i\hat p_j$. The gain in log likelihood is then simply the mutual information between $x_i$ and $x_j$. Thus, as long as no loop get closed by the procedure, the best candidate link corresponds to the pair of variables with maximum mutual information and the measure reads after $n$ steps $${\mathcal{P}}^{(n)}({{\mathbf{x}}}) = \prod_{(ij)\in{\mathcal{G}}^{(n)}} \frac{p_{ij}(x_i,x_j)}{p_i(x_i)p_j(x_j)} \prod_{i=1}^n
 p_i(x_i).$$ This suggests that a good initialization point for the algorithm is the maximum spanning tree with edges weights given by the relevant mutual information. This corresponds to the classical results of [@ChoLee] concerning inference using dependence trees. It is optimal in the class of singly connected graphical models. In the following, we will refer in the text to this specific 