${\wideparen{jk}}$ along ${\mathcal{T}}$ intersect on vertex $l$ as on Figure \[fig:vertex\].d, we obtain instead [^1] $$\chi_{ijkl} = 2\frac{3m_l^2-1}{1-m_l^2}\chi_{ij}\chi_{kl}.$$ For the situation corresponding to Figure \[fig:vertex\].e, we have $$p_(s_i,s_j,s_k,s_l) = \sum_{s_q}\frac{p(s_i,s_j,s_q)p(s_q,s_k,s_l)}{p(s_q)^2}$$ which leads to $$\chi_{ijkl} = 2\frac{3m_q^2-1}{1-m_q^2}\chi_{ij}\chi_{kl}.$$ Finally, for the situation corresponding to Figure \[fig:vertex\].f, we have $$p_(s_i,s_j,s_k,s_l) = \sum_{s_q}\frac{p(s_i,s_j)p(s_j,s_k,s_l)}{p(s_j)^2}$$ leading to $$\chi_{ijkl} = -\chi_{ik}\chi_{jl}-\chi_{jk}\chi_{il}+4\frac{m_km_q}{1-m_q^2}\chi_{ij}\chi_{lq}.$$

Sparse inverse estimation of covariance matrix
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Let us leave the Ising modeling issue aside for a while and introduce another related graph selection problem, named sparse inverse covariance estimation, which is defined on real continuous random variables. This method aims at constructing a sparse factor graph structure by identifying conditionally independent pairs of nodes in the graph, given empirical covariances of random variables. Assuming that all nodes in the graph follow a joint multi-variate Gaussian distribution with mean $\mu$ and covariance matrix $C$, the existing correlation between the nodes $i$ and $j$ given all the other nodes in the graph are indicated by the non-zero $ij$th entry of the precision matrix $C^{-1}$, while zero entries correspond to independent pairs of variables. Therefore, under the joint normal distribution assumption, selection of factor graph structures amounts to finding the sparse precision matrix that best describes the underlying 