# Fault Diagnosis in Partially Observable Petri Nets with Quantum Bayesian Learning
## 4 Case Analysis and Verification
### Convert the Conditional Probability Table to Quantum Probability Amplitude Table  
In the POPN model, there are unobservable transitions. When the observed transition sequence is given and the transition sequence is uncertain, we obtain the quantum probability amplitude table by Equations (1) and (6). In Figure 3, the uncertain path of \(t_{9}\) firing is \(t_{11}\to p_{9}\to t_{9}\), \(t_{7}\to p_{9}\to t_{9}\) and \(t_{10}\to p_{9}\to t_{9}\). And it is caused by the unobservable transition. In Figure 4, the uncertain path of \(t_{14}\) firing is \(t_{5}\to p_{7}\to t_{10}\to p_{14}\to t_{14}\), \(t_{6}\to p_{6}\to t_{8}\to p_{14}\to t_{14}\) and \(t_{5}\to p_{7}\to t_{7}\to p_{9}\to t_{9}\to p_{10}\to t_{8}\to p_{14}\to t_{14}\). Through the path superposition principle, the uncertainty of \(t_{8}\) and \(t_{10}\) leads to the uncertainty of transition \(t_{14}\) firing. In Figure 5, the uncertain path of \(t_{15}\) firing is \(t_{8}\to p_{14}\to t_{14}\to p_{11}\to t_{15}\), \(t_{10}\to p_{14}\to t_{14}\to p_{11}\to t_{15}\), \(t_{10}\to p_{11}\to t_{15}\) and. \(t_{8}\to p_{11}\to t_{15}\). The source of uncertainty is  
Figure 4: The transition \(t_{14}\) of the influence of quantum interference. Figure 3: The transition \(t_{9}\) of the influence of quantum interference. Figure 2: The QBPN model of fault transition \(t_{10}\). the unobservable transition \(t_{8}\), \(t_{10}\) and the observable transition \(t_{14}\). When the observable transition \(t_{14}\) is fired, the number of the uncertain paths is 2, and the reduction of path uncertainty leads to the reduction in the interference effect. The modification of the transition \(t_{14}\) firing probability is shown in Figure 6 as an example. When there is an uncertain path, in Table 1 the firing probability of \(t_{8}\) is 0.81 under condition \(t_{5}\) and quantum amplitude is \(\sqrt{0.81}e^{\theta_{1}}\). The firing probability of \(t_{10}\) is 0.19 under condition \(t_{5}\) and the quantum amplitude is \(\sqrt{0.19}e^{\theta_{2}}\). The firing probability of \(t_{14}\) is 0.63 under condition \(t_{5}\) and the quantum amplitude is \(\sqrt{0.63}e^{\theta_{3}}\). The firing probability of \(t_{14}\) is 0.43 under condition \(t_{10}\) and the quantum amplitude is \(\sqrt{0.43}e^{\theta_{4}}\). According to the probability information between transitions shown in Figure 6, the change curve of the probability value of quantum Bayes is obtained in Figure 7.  
We obtain the following formula:  
Figure 5: The transition \(t_{15}\) of the influence of quantum interference. Figure 6: Classical Bayesian probability and quantum Bayesian probability. \[Pr(t_{14}|t_{5})=\alpha\Big{|}\varphi_{t_{8}}\varphi_{t_{14}|t_{8}}+ \varphi_{t_{10}}\varphi_{t_{14}|t_{10}}\Big{|}^{2}=\alpha\bigg{(}\Big{|}\varphi_ {t_{8}}\varphi_{t_{14}|t_{8}}\Big{|}^{2}+\Big{|}\varphi_{t_{10}}\varphi_{t_{14}| t_{10}}\Big{|}^{2}\] \[+2\cdot\Big{|}\varphi_{t_{8}}\varphi_{t_{14}|t_{8}}\Big{|}\cdot \Big{|}\varphi_{t_{10}}\varphi_{t_{14}|t_{1  
In Figure 9, the more complicated case, \(t_{8}\) firing probability under condition \(t_{3}\). At this time, we adjust multiple interference angles to meet the demand. Possible paths: \(t_{3}\to t_{5}\to t_{7}\to t_{8}\), \(t_{3}\to t_{5}\to t_{10}\to t_{8}\), \(t_{3}\to t_{11}\to t_{7}\to t_{8}\) and \(t_{3}\to t_{11}\to t_{9}\to t_{8}\). Two of the different paths produce a superposition effect, and produce five interference angles \(\{\theta_{1},\theta_{2},\theta_{3},\theta_{4},\theta_{5}\}\). Fixing \(\{\theta_{3},\theta_{4},\theta_{5}\}\), we obtain the relation \(Pr(t_{8}|t_{3})\sim(\theta_{1},\theta_{2})\) between the firing probability of \(t_{8}\) under condition \(t_{3}\) and the interference angle \(\{\theta_{1},\theta_{2}\}\). In Figure 10, the probability transformation range is [0.83-0.88]. When \(\theta_{1}=0\wedge\theta_{2}=0\), we obtain the maximum, and when \(\theta_{1}=\pi\wedge\theta_{2}=\pi\), we obtain the minimum.