## Appendix B Additional Examples
\begin{matrix}\{a,c\}&\{a,d\}&\{b,c\}&\{b,d\}\\ e&\left(\begin{matrix}0.005+0.022=0.027&0.052+0.210=0.262&0.013+0.051=0.064&0.050+ 0.199=0.248\\ f&\left(\begin{matrix}0.006+0.089=0.153&0.037+0.051=0.087&0.040+0.056=0.096&0.026+0. 036=0.062\end{matrix}\right)\end{matrix}\end{matrix} \end{matrix}\]  
and we obtain the CPT for \(X_{3}\mid X_{1},X_{2}\) by normalising its columns as we did earlier with \(X_{4}\). **Example A2** (Composing and decomposing a CLGBN). : _Consider the CLGBN \(\mathcal{B}\) from Figure 3 top. The \(M=3\) discrete variables at the top of the network have the joint distribution below:_  
\[ \begin{matrix}\{X_{1},X_{2},X_{3}\}\\ \{a,c,e\}&\{b,c,e\}&\{a,d,e\}&\{b,d,e\}&\{a,c,f\}&\{b,c,f\}&\{a,d,f\}&\{b,d,f\} \\ \hline 0.040&0.036&0.040&0.084&0.160&0.144&0.160&0.336\end{matrix} \]  
_Its elements identify the components of the mixture that make up the global distribution of \(\mathcal{B}\), and the associated probabilities are the probabilities of those components._  
_We can then identify which parts of the local distributions of the \(N-M=3\) continuous variables (\(X_{4}\), \(X_{5}\) and \(X_{6}\)) we need to compute \(\mathrm{P}(X_{4},X_{5},X_{6}\mid X_{1},X_{2},X_{3})\) for each element of the mixture. The graphical structure of \(\mathcal{B}\) implies that \(\mathrm{P}(X_{4},X_{5},X_{6}\mid X_{1},X_{2},X_{3})=\mathrm{P}(X_{4},X_{5},X_{6 }\mid X_{2},X_{3})\) because the continuous nodes are \(d\)-separated from \(X_{1}\) by their parents. As a result, the following mixture components will share identical distributions which only depend on the configurations of \(X_{2}\) and \(X_{3}\):_  
\[\{a,c,e\},\{b,c,e\} \mapsto\{c,e\}, \{a,d,e\},\{b,d,e\} \mapsto\{d,e\},\] \[\{a,c,f\},\{b,c,f\} \mapsto\{c,f\}, \{a,d,f\},\{b,d,f\} \mapsto\{d,f\}.\]  
_For the mixture components with a distribution identified by \(\{c,e\}\), the relevant parts of the distributions of \(X_{4}\), \(X_{5}\) and \(X_{6}\) are:_  
\[X_{4} =0.1+0.2X_{5}+\varepsilon_{X_{4}}, \varepsilon_{X_{4}} \sim N(0,0.09);\] \[X_{5} =0.1+\varepsilon_{X_{5}}, \varepsilon_{X_{5}} \sim N(0,0.09);\] \[X_{6} =0.1+0.2X_{4}+\varepsilon_{X_{6}}, \varepsilon_{X_{6}} \sim N(0,1).\]  
_We can treat them as the local distributions in a GBN over \(\{X_{4},X_{5},X_{6}\}\) with a DAG equal to the subgraph of \(\mathcal{B}\) spanning only these nodes. If we follow the steps outlined in Section 3.2 and illustrated in Example 2, we obtain_  
\[ \begin{bmatrix}X_{4}\\ X_{5}\\ X_{6}\end{bmatrix} \sim N\left( \begin{bmatrix}0.120\\ 0.100\\ 0.124\end{bmatrix} ,\Sigma_{\{c,e\}}(\mathcal{B})= \begin{bmatrix}0.094&0.018&0.0 19\\ 0.018&0.090&0.004\\ 0.019&0.004&1.004\end{bmatrix} \right)\]  
_which is the multivariate normal distribution associated with the components \(\{a,c,e\}\) and \(\{b,c,e\}\) in the mixture. Similarly, the relevant parts of the distributions of \(X_{4}\), \(X_{5}\) and \(X_{6}\) for \(\{d,e\}\) are_  
\[X_{4} =0.6+0.8X_{5}+\varepsilon_{X_{4}}, \varepsilon_{X_{4}} \sim N(0,0.36);\] \[X_{5} =0.2+\varepsilon_{X_{5}}, \varepsilon_{X_{5}} \sim N(0,0.36);\] \[X_{6} =0.1+0.2X_{4}+\varepsilon_{X_{6}}, \varepsilon_{X_{6}} \sim N(0,1);\]  
_and jointly_  
\[ \begin{bmatrix}X_{4}\\ X_{5}\\ X_{6}\end{bmatrix} \sim N\left( \begin{bmatrix}0.760\\ 0.200\\ 0.252\end{bmatrix} ,\Sigma_{\{d,e\}}(\mathcal{B})= \begin{bmatrix}0.590&0.288&0. 118\\ 0.288&0.360&0.058\\ 0.118&0.058&1.024\end{bmatrix} \right)\]  
_for the components \(\{a,d,e\}\) and \(\{b,d,e\}\). For the components \(\{a,c,f\}\) and \(\{b,c,f\}\), the local distributions identified by \(\{c,f\}\) are_  
\[X_{4} =0.1+0.2X_{5}+\varepsilon_{X_{4}}, \varepsilon_{X_{4}} \sim N(0,0.09);\] \[X_{5} =0.4+\varepsilon_{X_{5}}, \varepsilon_{X_{5}} \sim N(0,0.81);\] \[X_{6} =0.1+0.2X_{4}+\varepsilon_{X_{6}}, \varepsilon_{X_{6}} \sim N(0,1);\]  
_and the joint distribution of \(X_{4}\), \(X_{5}\) and \(X_{6}\) is_  
\[ \begin{bmatrix}X_{4}\\ X_{5}\\ X_{6}\end{bmatrix} \sim N\left( \begin{bmatrix}0.180\\ 0.400\\ 0.136\end{bmatrix} ,\Sigma_{\{c,f\}}(\mathcal{B})=