## 5 A Case Study
### Evaluating the Reliability of Servo Turret  
As mentioned above, we obtained two failures in the reliability test and could regard others that did not fail as right-censored data. The likelihood function under Weibull distribution is as follows:  
\[L(t\mid\alpha,\beta)=\prod_{i=1}^{n}\left[\frac{\alpha}{\beta}\left(\frac{t_{i} }{\beta}\right)^{\alpha-1}e^{-\left(\frac{t}{\beta}\right)}\right]^{\alpha} \left[e^{-\left(\frac{t}{\beta}\right)^{\alpha}}\right]^{1-\delta} \tag{14}\]  
where \(\alpha\) is the shape parameter, \(\beta\) is the scale parameter, \(t_{i}\) is the failure time or censored time and \(\delta_{i}\) is the censoring indicator; if the \(i\)th data is censored, \(\delta_{i}=0\), and \(\delta_{i}=1\) if it failed. Taking derivatives of Equation (15) with respect to the parameters \(\alpha\) and \(\beta\), the following log-likelihood equations are obtained as shown in Equation (16). Thus, the shape parameter \(\alpha=5927.18\) and shape parameter \(\beta=1.00\). \[\left\{ \begin{array}{c}\frac{\partial\ln L(t|\alpha,\beta|)}{\partial\alpha} =\sum_{i=1}^{n}\left\{\delta_{i}\left[\frac{1}{\alpha}+\ln\frac{t_{i}}{\beta} -\left(\frac{t_{i}}{\beta}\right)^{\alpha}\left(\ln\frac{t_{i}}{\beta}\right) \right]+\left(1-\delta_{i}\right)\left[-\left(\frac{t_{i}}{\beta}\right)^{ \alpha}\left(\ln\frac{t_{i}}{\beta}\right)\right]\right\}\\ \frac{\partial\ln L(t|\alpha,\beta|)}{\partial\beta}=\sum_{i=1}^{n}\left\{ \delta_{i}\left[-\frac{\alpha}{\beta}+\alpha\frac{t_{i}^{\alpha}}{\beta^{ \alpha+1}}\right]+\left(1-\delta_{i}\right)\left(\alpha\frac{t_{i}^{\alpha}}{ \beta^{\alpha+1}}\right)\right\}\end{array} \right. \tag{15}\]  
Based on the above, the prior of the system using limited failure data could be obtained. The posterior could be evaluated by the ABM method combined with experts' judgements. First, the induced system prior was evaluated based on the system structure, and as the servo turret was a series system, the induced system reliability model was \(\theta_{\text{sys}}=\prod_{i=1}^{n}\theta_{i}\), where \(\theta_{i}\) is the subsystem reliability distribution. To reflect the need to use ABM, the different weights were compared as shown in Figure 5. It can be seen that the results vary significantly with the pooling weight. In this case, the subsystem prior from the experts and system prior system from the reliability test were used for reliability evaluation, and these priors were collected from different domains. In this case, the subsystem prior information from experts and system prior information from the reliability test were available. It was necessary to aggregate the inconsistent information induced through the expert system and the data-driven observation function. Since this information was collected from different domains, a suitable prior of pooling weights needed to be determined to reduce the error in the uncertainty quantification. Figure 4: The weight-directed graph for different criteria. Different experts' judgements were aggregated by the mean and ABM, as proposed by this paper and shown in Figure 6. In practice, it seems reasonable to perform aggregation using a simple equal-weight weighted average; because of the many uncertainties in the assessment, it is difficult to defend the difference in expert weights. However, five experts were invited to estimate the RIF in this paper and covered a very wide range, and the weights assigned as a measure of confidence in the estimates differed greatly. The approach proposed for reliability evaluation in this paper balanced the judgment of these experts and the limited reliability of the test results, obtaining the result by probabilistic modeling and avoiding subjective weight allocation. Finally, the proposed framework was used to evaluate the posterior. The number of original sampled importance weights was set to \(10^{4}\), the number of selected weights was assigned as \(10^{4}\), and the percentage parameter for determining the threshold weight value was 90%.