Algorithm 1 Extended VMP (Mean-field assumption)

Require:  $G=(V,E)$, $h_{1:M+L}$, $N_{iterations}$, $q_{initial}$  for  $j=1\cdots M+L$  do   initialize posteriors $q\left(h_j\right)$ using $q_{initial}$  end for  for i = 1...$N_{iterations}$ do   Set Free Energy $\mathcal{F}=0$   for  $j=1\cdots M+L$  do    Calculate messages ${\overrightarrow{m}}_{h_j}\left(h_j\right)$ and ${\overleftarrow{m}}_{h_j}\left(h_j\right)$ using Section 3.6    Calculate posterior $q\left(h_j\right)$ using Section 3.5    if  $h_j\in z_{1:M}$  then     Calculate entropy ${\mathcal{H}}_{h_j}$ using Section 3.7     Update Free Energy $\mathcal{F}=\mathcal{F}-{\mathcal{H}}_{h_j}$    end if   end for   for all  $v\in V_f$  do    Calculate energy ${\mathcal{U}}_v$ using Section 3.7    Update Free Energy $\mathcal{F}=\mathcal{F}+{\mathcal{U}}_v$   end for   return $q\left(h\right)$ for all $h\in h_{1:M+L}$ and $\mathcal{F}$  end for