# Gaussian Mixture Reduction for Time-Constrained Approximate Inference in Hybrid Bayesian Networks +
## 2 Preliminaries
### Hybrid Message Passing Inference  
#### 2.1.1 Structure of Hybrid Bayesian Network  
A general hybrid BN can contain both discrete and continuous nodes. A node in a hybrid BN can be categorized according to its type (i.e., discrete or continuous), its parent node type(s) (i.e., discrete, continuous or hybrid with at least one discrete and one continuous node) and its child node type(s) (i.e., discrete, continuous or hybrid). The following table shows all possible classifications of nodes in a hybrid BN (D stands for discrete; C stands for continuous; and H stands for hybrid). As shown in Table 1, there are 18 node categories in a general hybrid BN. Various special cases impose restrictions eliminating some of the 18 categories. A hybrid BN in which no discrete node may have a continuous parent node is called a conditional hybrid BN [29]. That is, a conditional hybrid BN may contain Types 1, 2, 3, 11, 14 and 17 from Table 1. These six cases are shown in Figure 1. In the figure, a rectangle indicates a discrete node and a circle indicates a continuous node. For example, Type 1 has a discrete node \(B\) with its discrete parent node \(A\) and discrete child node \(C\), while Type 3 has a discrete node \(B\) with its discrete parent node \(A\) and hybrid child nodes \(C\) and \(Y\). A general hybrid BN places no restriction on the type of probability distribution for a continuous node. If all continuous nodes in a hybrid BN have Gaussian probability distributions, the BN is called Gaussian hybrid BN. A BN that is both a conditional hybrid BN and a Gaussian hybrid BN is called a conditional Gaussian (CG) BN. CG BNs can be further classified into two sub-categories: conditional linear Gaussian (CLG) BNs and conditional nonlinear Gaussian (CNG) BNs. For the CLG BNs, the Gaussian conditional distributions are always linear functions of the Gaussian parents. That is, if \(X\) is a continuous node \(X\) with \(n\) continuous parents \(U_{1}\),..., \(U_{n}\) and \(m\) discrete parents \(A_{1}\),..., \(A_{m}\), then the conditional distribution \(\text{p}(X\mid\textbf{{u}},\textbf{{a}})\) given parent states \(\textbf{{U}}\)=\(\textbf{{u}}\) and \(\textbf{{A}}\)=\(\textbf{{a}}\) has the following form:  
\[p(X|\textbf{{u}},\textbf{{a}})=N(L^{(\textbf{{a}})}(\textbf{{u}}),\sigma^{( \textbf{{a}})}), \tag{1}\]  
where \(L^{(\textbf{{a}})}(\textbf{{u}})=m^{(\textbf{{a}})}+b_{1}{}^{(\textbf{{a}})}u _{1}+\ldots+b_{n}{}^{(\textbf{{a}})}u_{n}\) is a linear function of the continuous parents, with intercept \(m^{(\textbf{{a}})}\), coefficients \(b_{i}{}^{(\textbf{{a}})}\) and standard deviation \(\sigma^{(\textbf{{a}})}\) that depend on the state \(a\) of the discrete parents. A Gaussian conditional distribution for a continuous node in a CNG BN can be any function of the Gaussian and discrete parents. The form is similar to Equation (1) except that \(L^{(\textbf{{a}})}(\textbf{{u}})\) can be a nonlinear function. Note that the types of Figure 1 cover any number of parent and child nodes. Thus, the discrete node B can have a set of discrete parent nodes (i.e., **A** = {\(A_{1}\), \(A_{2}\),..., \(A_{l}\)}), a set of discrete child nodes (i.e., C = {\(C_{1}\), \(C_{2}\),..., \(C_{m}\)}) and/or a set of continuous child nodes (i.e., **Y** = {\(Y_{1}\), \(Y_{2}\),..., \(Y_{n}\)}). The continuous node \(X\) can have a set of discrete parent nodes (i.e., **A** = {\(A_{1}\), \(A_{2}\),..., \(A_{l}\)}), a set of \begin{table}
\begin{tabular}{c c c c c c c c c} \hline _Parent Node Type(s)_ & & & **D** & & & **C** & & & **H** & \\ _Child Node Type(s)_ & **D** & **C** & **H** & **D** & **C** & **H** & **D** & **C** & **H** \\ \hline
**D** & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
**C** & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline \end{tabular}
\end{table} 
Table 1: Possible Node Types in a Hybrid Bayesian Network (BN). continuous parent nodes (i.e., \(\mathbf{U}=\{U_{1},U_{2},\ldots,U_{m}\}\)) and a set of continuous child nodes (i.e., \(\mathbf{Y}=\{Y_{1},Y_{2},\)\(\ldots,Y_{n}\}\)).