## 2 Background
### Bayesian Networks  
The BN (referred to as belief networks) represents a hypothesis of rationalizing from uncertain evidence to uncertain conclusions, since it can perform the factorization of the collective distribution of variables, based on the conditional dependencies. BN is helpful in addressing uncertainty and incompleteness problems; thus, it is extensively applied in several domains. BN graphically depicts the logical associations between variables and recognizes the connections between these variables by conditional probabilities. By interpretation, a BN represents a directed acyclic graph (DAG), which encodes a conditional probability distribution. Nodes and arcs are vital components of BN, the nodes symbolize arbitrary variables and the arcs signify random relations between variables. There is a probability function for each state of the node, and conditional probabilities are used to exhibit the associations between variables. BNs are probabilistic graphical models; these visual structures characterize the information about an uncertain system [17]. BNs are generally utilized for examining the hazards and vulnerabilities of networks, which are acyclic graphs that provide a quantitative and qualitative assessment of risks. [PERSON] initially proposed the BN-based approach in 1985 and was usually utilized to distribute random information in AI. Owing to the unique functionality of BN for constructing the structures and algorithms, it is successfully used in e-commerce, transportation, data mining, energy control, etc. It is a DAG-based probability rationalization and appropriate for uncertainty representation of queries. BN must be a DAG and CPT (conditional probability table). BN has been demonstrated to be a powerful tool for solving several problems with uncertain knowledge illustration and reasoning [18; 19; 20]. The BN formula is represented in Equation (1):  
\[\mathrm{P}\big{(}\mathrm{X}_{\mathrm{j}}\,|\,\mathrm{Y}\big{)}=\frac{\mathrm{P} \Big{(}\frac{\mathrm{Y}}{\mathrm{X}_{\mathrm{j}}}\Big{)}\mathrm{P}\big{(} \mathrm{X}_{\mathrm{j}}\big{)}}{\sum_{\mathrm{j=1}}^{\mathrm{m}}\mathrm{P} \Big{(}\frac{\mathrm{Y}}{\mathrm{X}_{\mathrm{j}}}\Big{)}\mathrm{P}\big{(} \mathrm{X}_{\mathrm{j}}\big{)}} \tag{1}\]  
where \(\mathrm{P}(\cdot\mid\,. )\) stands for the conditional probability distribution. Suppose the sample space \(\mathrm{N}\) of experiment \(\mathrm{L}\), \"\(\mathrm{Y}\)\" is the random event of \(\mathrm{L}\). \(\mathrm{X}_{\mathrm{1}}\), \(\mathrm{X}_{\mathrm{2}}\),..., \(\mathrm{X}_{\mathrm{n}}\) is the incompatible set of possibilities in experiment \(\mathrm{L}\), and \"\(\mathrm{X}_{\mathrm{j}}\)\" represents the entire group event from (\(\mathrm{j=1,2,\ldots,m}\)). Figure 1 represents the three-variable examples of BN structure. A BN compremetsh two types of nodes, i.e., the parent and child nodes. The parent node (cause) is at the start of any directed edge; the child node (fruit) is at the end. The directed edge specifies that the two nodes are interrelated. In Figure 1, \(\mathrm{X}\), \(\mathrm{Y}\) are the two-parent nodes of \(\mathrm{Z}\). \(\mathrm{Z}\) is the child nodes of \(\mathrm{X}\) and \(\mathrm{Y}\). Prior probability: \(\mathrm{P}(\mathrm{X})\) characterizes the probability of event \(\mathrm{X}\); \(\mathrm{P}(\mathrm{Y})\) is the probability of event \(\mathrm{Y}\); \(\mathrm{P}(\mathrm{Z}\!\mid\mathrm{X},\mathrm{Y})\) is the probability that the event \(\mathrm{Z}\) occurs before the condition that occurs at \(\mathrm{X}\) and \(\mathrm{Y}\). The posterior probability, \(\mathrm{P}(\mathrm{X}\!\mid\mathrm{Z})\), \(\mathrm{P}(\mathrm{Y}\!\mid\mathrm{Z})\), and so on, can be obtained through the known prior probability. A node without a parent is known as a root node, and a node without children is termed as a leaf node. In BNs, nodes with links represent system variables demonstrating uncertain dependencies.