# BN-RA: A Hybrid Model for Risk Analysis of Overload-Induced Early Cable Fires
## 3 Construction of a BN
### Risk BN for Overload-Induced Early Cable Fires  
BNs were introduced by Dr. [PERSON]. A BN captures conditional probability relationships based on the Bayesian theorem, and connects the uncertainty relationships  
Figure 1: Risk transmission paths of overload-induced cable fires.  
among different events by a directed acyclic graph, thus yielding a causal probability network that contains the transmission paths of events. In Figure 2, a sample BN is shown, where three layers of nodes are connected with directed acyclic arrows. The nodes are considered the model variables, and relationships between variables are represented by directed acyclic arrows. Any sub-node of a given node is called a child node, and the given node is the parent node. Root nodes have no links directed toward them (i.e., they do not have parent nodes), and represent basic events whose probabilities are assigned as prior probabilities. These probabilities are obtained from standard sources of failure data or based on expert elicitation in the case of unavailable failure data. Nodes with no child nodes are called leaf nodes, and are quantified using a conditional probability table (CPT). Both intermediate and leaf nodes are quantified using a CPT, considering the causal dependence between them [42].  
In a BN, according to the inference principle of Bayesian decision theory, the conditional probability of random events \(X\) and \(Y\) is given by:  
\[P(X|Y)=\frac{P(Y|X)P(X)}{P(Y)} \tag{1}\]  
where \(P(X|Y)\) is the probability of occurrence of \(X\) when \(Y\) occurs.  
If event \(X_{i}\) leads to event \(Y\), \(Y\) is regarded as a child node. After the evidence associated with the child node \(Y\) is updated, the posterior probability of the parent node \(X_{i}\) is written as:  
\[P(X_{i}|Y)=\frac{P(Y|X_{i})P(X_{i})}{\sum_{i=1}^{n}P(Y|X_{i})P(X_{i})},\;i=1, \;2,\;3,\;\ldots,\;n \tag{2}\]  
If \(X_{i}\) (\(i=1,\;2,\;\ldots,\;n\)), \(n\) events occur, then the occurrence probability of these events is given by the joint probability  
\[P(Y)=P(Y|X_{1})P(X_{1})+P(Y|X_{2})P(X_{2})+\cdots+P(Y|X_{n})P(X_{n}) \tag{3}\]  
A BN can be used as a risk analysis model for complex systems [43]. Risk factors are represented as nodes in the BN, and risk transmission is encoded in the directed acyclic graph, yielding a causal probability network that represents the risk evolution process. By calculating conditional probability, the probability of event occurrence and possible consequences can be measured quantitatively. At the same time, a BN can integrate quantitative information from different channels, can deal with continuity and multi-state variables, and can carry out prediction and diagnostic analyses.  
To establish an overload-induced early cable fire risk BN model, it is necessary to determine the network structure and parameters. Then, the model BN can be established  
Figure 2: Basic structure of a BN.  
according to the early evolution stage of the fire risk transmission paths, as shown in Figure 1. The established BN structure with node states is shown in Figure 3.