## 3 Common Distributional Assumptions for Bayesian Networks
### Gaussian BNs  
_Gaussian BNs_[35] (GBNs) model \(\mathbf{X}\) with a multivariate normal random variable \(N(\boldsymbol{\mu}_{\mathcal{B}},\)\(\Sigma_{\mathcal{B}})\) and assume that the \(X_{i}\) are univariate normals linked by linear dependencies,  
\[X_{i}\mid\Pi_{X_{i}}\sim N(\mu_{X_{i}}+\Pi_{X_{i}}\boldsymbol{\beta}_{X_{i}}, \sigma_{X_{i}}^{2}), \tag{3}\]  
which can be equivalently written as linear regression models of the form  
\[X_{i}=\mu_{X_{i}}+\Pi_{X_{i}}\boldsymbol{\beta}_{X_{i}}+\varepsilon_{X_{i}}, \varepsilon_{X_{i}}\sim N(0,\sigma_{X_{i}}^{2}). \tag{4}\]  
The parameters in (3) and (4) are the regression coefficients \(\boldsymbol{\beta}_{X_{i}}\) associated with the parents \(\Pi_{X_{i}}\), an intercept term \(\mu_{X_{i}}\) and the variance \(\sigma_{X_{i}}^{2}\). They are usually estimated by maximum likelihood, but Bayesian and regularised estimators are available as well [1]. The link between the parameterisation of the global distribution of a GBN and that of its local distributions is detailed in Pourahmadi [36]. We summarise it here for later use. * _Composing the global distribution._ We can create an \(N\times N\) lower triangular matrix \(C_{\mathcal{B}}\) from the regression coefficients in the local distributions such that \(C_{\mathcal{B}}C_{\mathcal{B}}^{\mathrm{T}}\) gives \(\Sigma_{\mathcal{B}}\) after rearranging rows and columns. In particular, we: 1. Arrange the nodes of \(\mathcal{B}\) in the (partial) topological ordering induced by \(\mathcal{G}\), denoted \(X_{(i)},i=1,\ldots,N\). 2. The \(i\)th row of \(C_{\mathcal{B}}\) (denoted \(C_{\mathcal{B}}[i;\,\boldsymbol{\cdot}\,]\), \(i=1,\ldots,N\)) is associated with \(X_{(i)}\). We compute its elements from the parameters of \(X_{(i)}\mid\Pi_{X_{(i)}}\) as \[C_{\mathcal{B}}[i;i]=\sqrt{\sigma_{X_{(i)}}^{2}}\qquad\qquad\text{and}\qquad \qquad C_{\mathcal{B}}[i;\,\boldsymbol{\cdot}\,]=\boldsymbol{\beta}_{X_{(i)}} C_{\mathcal{B}}[\Pi_{X_{(i)}};\,\boldsymbol{\cdot}\,],\] where \(C_{\mathcal{B}}[\Pi_{X_{(i)}};\,\boldsymbol{\cdot}\,]\) are the rows of \(C_{\mathcal{B}}\) that correspond to the parents of \(X_{(i)}\). The rows of \(C_{\mathcal{B}}\) are filled following the topological ordering of the BN. 3. Compute \(\widetilde{\Sigma}_{\mathcal{B}}=C_{\mathcal{B}}C_{\mathcal{B}}^{\mathrm{T}}\). 4. Rearrange the rows and columns of \(\widetilde{\Sigma}_{\mathcal{B}}\) to obtain \(\Sigma_{\mathcal{B}}\). Intuitively, we construct \(C_{\mathcal{B}}\) by propagating the node variances along the paths in \(\mathcal{G}\) while combining them with the regression coefficients, which are functions of the correlations between adjacent nodes. As a result, \(C_{\mathcal{B}}C_{\mathcal{B}}^{\mathrm{T}}\) gives \(\Sigma_{\mathcal{B}}\) after rearranging the rows and columns to follow the original ordering of the nodes. The elements of the mean vector \(\boldsymbol{\mu}_{\mathcal{B}}\) are similarly computed as \(\mathrm{E}(X_{(i)})=\Pi_{X_{(i)}}\boldsymbol{\beta}_{X_{(i)}}\) iterating over the variables in topological order. * _Decomposing the global distribution._ Conversely, we can derive the matrix \(C_{\mathcal{B}}\) from \(\Sigma_{\mathcal{B}}\) by reordering its rows and columns to follow the topological ordering of the variables in \(\mathcal{G}\) and computing its Cholesky decomposition. Then \[R=\mathrm{I}_{N}-\mathrm{diag}(C_{\mathcal{B}})C_{\mathcal{B}}^{-1},\] contains the regression coefficients \(\boldsymbol{\beta}_{X_{(i)}}\) in the elements corresponding to \(X_{(i)},\Pi_{X_{(i)}}\) (Here \(\mathrm{diag}(C_{\mathcal{B}})\) is a diagonal matrix with the same diagonal elements as \(C_{\mathcal{B}}\) and \(\mathrm{I}_{N}\) is the identity matrix.) Finally, we compute the intercepts \(\mu_{X_{i}}\) as \(\boldsymbol{\mu}_{\mathcal{B}}-R\boldsymbol{\mu}_{\mathcal{B}}\) by reversing the equations we used to construct \(\boldsymbol{\mu}_{\mathcal{B}}\) above.