## 2 Hybrid CM-Enabled DBN Method
### Dynamic Bayesian Network  
A BN is a type of probabilistic reasoning network based on Bayesian conditional probability and graph theory [32]. A BN assumes the form of a directed acyclic graph, as visually illustrated in Figure 2, wherein nodes symbolize random variables, and directed arcs delineate the causal relationships among these variables. In this context, a node responsible for generating an arc is denoted as the parent node, while the node receiving the directed arc is referred to as the child node [33]. The conditional dependencies between child nodes and their parent nodes were characterized using conditional probability tables (CPTs). Considering the conditional dependencies of variables, the joint probability distribution \(P(X)\) of variable \(X=\{X_{1},\ldots,X_{n}\}\) in the BN can be expressed as follows:  
\[P(X)=\prod_{i=1}^{n}P(X_{i}|Pa(X_{i})) \tag{1}\]  
where \(Pa(X_{i})\) indicates the parent set of \(X_{i}\) for any \(i=1,\ldots,n\); \(P(X_{i}|Pa(X_{i}))\) represents the conditional probability of variable \(X_{i}\), and the chain rule of probability and the property of conditional independence have been extensively adopted [34].  
Compared to the traditional static BN, the DBN is a derivative of the BN, which introduces the influence of the internal time series. The dependencies between nodes in a DBN can be distinguished into two types: (i) conditional dependencies under the same time slice and (ii) conditional dependencies across different time slices [22]. A DBN can be defined within a double variable (\(B_{0}\), \(B_{\rightarrow}\)) framework, where \(B_{0}\) is a BN that defines the probability distribution P(\(X_{0}\)) at the initial time, and \(B_{\rightarrow}\) is called a \"two-slice temporal Bayesian network (2 TBN),\" which defines the conditional distribution among variables between two time slices according to the following equation [35;36]:  
\[P(X_{t}|X_{t-1})=\prod_{i=1}^{n}P(X_{i,t}|Pa(X_{i,t})) \tag{2}\]  
where \(X_{i,t}\) denotes the \(i\)-th node at time \(t\), \(Pa(X_{i,t})\) indicates the parent node of \(X_{i,t}\) for the previous and same time slices, and \(n\) indicates the number of nodes in the network.  
Meanwhile, a DBN typically follows two assumptions: (1) the Markov hypothesis, which is the probability that the nodes in time \(t\) are only affected by time \(t-1\), independent of the time segment before \(t-1\); and (2) the stationary hypothesis, which states that the conditional probability in \(B_{\rightarrow}\) does not change with time [37]. According to the initial probability distribution in \(B_{0}\) and the conditional distribution between adjacent time slices in \(B_{\rightarrow}\), the joint probability distribution of the nodes from time \(t=0\) to \(t=T\) can be expressed as follows:  
\[P(X_{0:T})=\prod_{t=0}^{T}\prod_{i=1}^{n}P(X_{i,t}|Pa(X_{i,t})) \tag{3}\]  
where \(X_{0:T}\) represents all nodes from time slice \(t=0\) to \(t=T\).  
Figure 3a presents the initial network \(B_{0}\) of a DBN, where all variables are static. Figure 3b depicts 2 TBN \(B_{\rightarrow}\), where the virtual arcs represent the conditional dependencies  
Figure 2: Simple Bayesian network.  
of the nodes under different time slices. Extending \(B_{\rightarrow}\) to a certain number of time slices yields a complete DBN. To simplify the structure, an abstract form of the DBN is generally used [38], as shown in Figure 3c. The dotted lines in the image refer to the corresponding nodes in the preceding and following time periods.