## 2 Coupling Analysis of Risk Factors
### Probability Assessment of the Coupled Risk Factors Based on the BN  
The Bayesian network is mainly applied to evaluate the probability of the occurrence of coupled risk factors, with the specific process as follows. 1. BN modeling Based on the Direct-Relation Matrix in the DEMATEL Model  
Although a BN utilizes a directed graph, while DEMATEL technology uses a direct-relation matrix, they are both effective tools for describing the direct causal relationship between factors. The direct-relation matrix can be utilized to construct the initial causal diagram, in which the possible circulations can be eliminated to obtain the modified BN diagram. A threshold value for the direct-relation matrix is determined based on expert judgment, and the elements that are greater than the threshold value in the direct-relation matrix can be converted as valid nodes and arcs in the initial causal diagram [12]. In view of the possible circulations in the initial causal diagram, it is necessary to identify and eliminate the circulations based on the following principles in order to transform it to a BN. 1. The BN should reflect not only the causal relationship between factors, but also the directions. There may be circulations in the initial causal diagram due to vague concepts of correlation and causality. For instance, there is a strong causal relation in \(\alpha\rightarrow\beta\), but with no relation in \(\beta\rightarrow\alpha\). 2. Since the BN mainly represents the direct causal relationship, there may be circulations in the initial causal diagram due to the indirect causality. 3. There may be circulations in the initial causal diagram because of time or other variables that exert an influence on the causal relationship between factors. The above principles are adopted to check the bidirectional arc and closed loop arc in the initial causal diagram, so as to identify and eliminate the cycle arcs to obtain a Bayesian network diagram. 2. The Parameterization of BN  
The diagrams of the Bayesian network are visual descriptions of the factorization of joint probability distributions [38], where nodes represent risk factors, and arcs represent probability dependencies between factors. For the reason that a diagram can depict the structure of a probability domain but cannot supply numerical characteristics, the conditional probability table (CPT) of each node needs to be established to parameterize the BN. When two nodes \(\alpha\) and \(\beta\) are directly connected, such as \(\alpha\rightarrow\beta\), \(\alpha\) and \(\beta\) are viewed as parent node and child node, respectively. When performing the risk assessment based on a BN, the obtained risk factors as well as a \"risk\" event are defined as nodes with two states of occurrence and non-occurrence. The prior probability of all nodes and the conditional probability under the combination of all of the states of these nodes and their parent nodes are input into the CPT to complete the parameterization. The calculation of the prior probability and conditional probability of each node is as follows. According to the classification of risk factors in Section 3.2, each node in the Bayesian network is a set of multiple concrete risk factors. If the node \(F_{f}\) contains risk factors \(Q_{1},Q_{2}\cdots Q_{t}\), the probability of each factor is \(P_{1},P_{2}\cdots P_{t}\). The prior probability of node \(F_{f}\) is defined as \(\mathrm{P}\big{(}F_{f}\big{)}\), and the formula is as follows on the assumption that the factors are independent of each other. \[\mathrm{P}\big{(}F_{f}\big{)}=\sum\
olimits_{i=1}^{t}P_{i}-\sum\
olimits_{i= 2}^{t}P_{1}P_{i}-\sum\
olimits_{i=3}^{t}P_{2}P_{i}-\cdots-P_{t-1}P_{t}. \tag{4}\]  
When the node \(F_{i}\) is conditional on its parent node \(F_{d}\), the probability is defined as \(\mathrm{P}(F_{i}|F_{d})\), representing the probability of the occurrence of node \(F_{i}\) when the parent node \(F_{d}\) occurs.