## 3 Common Distributional Assumptions for Bayesian Networks
### Conditional Linear Gaussian BNs  
Finally, _conditional linear Gaussian BNs_[37] (CLGBNs) subsume discrete BNs and GBNs as particular cases by combining discrete and continuous random variables in a mixture model. If we denote the former with \(\mathbf{X}_{D}\) and the latter with \(\mathbf{X}_{G}\), so that \(\mathbf{X}=\mathbf{X}_{D}\cup\mathbf{X}_{G}\), then:  
* Discrete \(X_{i}\in\mathbf{X}_{D}\) are only allowed to have discrete parents (denoted \(\Delta_{X_{i}}\)), and are assumed to follow a multinomial distribution parameterised with CPTs. We can estimate their parameters in the same way as those in a discrete BN.  
Figure 1: DAGs and local distributions for the GBNs \(\mathcal{B}\) (**top**) and \(\mathcal{B}^{\prime}\) (**bottom**) used in Examples 2 and 6–9.  
* Continuous \(X_{i}\in\mathbf{X}_{G}\) are allowed to have both discrete and continuous parents (denoted \(\Gamma_{X_{i}}\), \(\Delta_{X_{i}}\cup\Gamma_{X_{i}}=\Pi_{X_{i}}\)). Their local distributions are \[X_{i}\mid\Pi_{X_{i}}\sim N\Big{(}\mu_{X_{i},\beta_{X_{i}}}+\Gamma_{X_{i}}\beta_ {X_{i},\delta_{X_{i}}},\sigma^{2}_{X_{i},\delta_{X_{i}}}\Big{)},\] which is equivalent to a mixture of linear regressions against the continuous parents with one component for each configuration \(\delta_{X_{i}}\in\mathit{Val}(\Delta_{X_{i}})\) of the discrete parents: \[X_{i}=\mu_{X_{i},\delta_{X_{i}}}+\Gamma_{X_{i}}\beta_{X_{i},\delta_{X_{i}}}+ \varepsilon_{X_{i},\delta_{X_{i}}},\qquad\qquad\varepsilon_{X_{i},\delta_{X_{ i}}}\sim N\Big{(}0,\sigma^{2}_{X_{i},\delta_{X_{i}}}\Big{)}.\] If \(X_{i}\) has no discrete parents, the mixture reverts to a single linear regression like that in (4). The parameters of these local distributions are usually estimated by maximum likelihood like those in a GBN; we have used hierarchical regressions with random effects in our recent work [38] for this purpose as well. Bayesian and regularised estimators are also an option [5]. If the CLGBN comprises \(|\mathbf{X}_{D}|=M\) discrete nodes and \(|\mathbf{X}_{G}|=N-M\) continuous nodes, these distributional assumptions imply the partial topological ordering \[\underbrace{\Big{\{}X_{(1)},\ldots,X_{(M)}\Big{\}}}_{\text{discrete nodes}},\, \underbrace{\Big{\{}X_{(M+1)},\ldots,X_{(N)}\Big{\}}}_{\text{continuous nodes}}.\] (5) The discrete nodes jointly follow a multinomial distribution, effectively forming a discrete BN. The continuous nodes jointly follow a multivariate normal distribution, parameterised as a GBN, for each configuration of the discrete nodes. Therefore, the global distribution is a Gaussian mixture in which the discrete nodes identify the components, and the continuous nodes determine their distribution. The practical link between the global and local distributions follows directly from Sections 3.1 and 3.2.  
**Example 3** (Composing and decomposing a CLGBN).: _For reasons of space, this example is presented as Example A2 in Appendix B._  
The complexity of composing and decomposing the global distribution is then  
\[\underbrace{\mathcal{O}\Big{(}M^{M}\Big{)}}_{\text{convert between CFIs and component probabilities}}+\underbrace{\mathcal{O}\Big{(}(N-M)^{3}\big{)}^{\mathit{Val}(\Delta)}}_{\text{(de) compose the distinct component distributions}}\]  
where \(\Delta=\bigcup_{X_{i}\in\mathbf{X}_{G}}\Delta_{X_{i}}\) are the discrete parents of the continuous nodes.