## 3 Case Study
### Results and Discussion  
(1) Convergence monitoring  
This study chose the Gelman-Rubin method [30] to monitor convergence for model parameters on Markov chains. For an iterative simulation with \(m\) sequences (\(m\) equal 2 in  
Figure 4: Sampling results of the improved Weibull parameters. the case study) of length \(n\), the convergence can be monitored via the indicator of potential scale reduction estimated by the following equations:  
\[B=\frac{n}{m-1}\sum_{i=1}^{m}\left(\overline{x}_{i.}-\overline{x}_{-}\right)^{2} \tag{25}\]  
\[W=\frac{1}{m}\sum_{i=1}^{m}\left[\frac{1}{n-1}\sum_{j=1}^{n}\left(\overline{x} _{ij}-\overline{x}_{i.}\right)^{2}\right] \tag{26}\]  
\[\hat{R}=\sqrt{\frac{\sqrt{Var(x|y)}}{W}}=\sqrt{\frac{\frac{n-1}{n}W+\frac{1}{n }B}{W}} \tag{27}\]  
where \(B\) denotes the variance between the \(m\) sequence means; \(W\) denotes the average of the \(m\) within-sequence variances; \(x\) is the estimated model parameter; \(Var(x\,|\,y)\) represents the current variance estimate of the marginal posterior distribution of \(x\); \(\hat{R}\) denotes the potential scale reduction. When \(n\) approaches infinity and Markov chains converge, the variance estimate \(Var(x\,|\,y)\) and within-sequence variance will approach the true variance of parameter \(x\). Therefore, the convergence of estimated parameters can be judged to be adequate when \(\hat{R}\) nears 1. Figure 5 shows the 94% high density intervals (HDI) of all model parameters on the left, and the estimated values of potential scale reduction on the right, from which it allows concluding that all model parameters have adequate convergence on Markov chains. Figure 5: Gelman-Rubin convergence monitoring results. Model and sampling evaluation  
In this case study, Bayesian Fraction of Missing Information (BFMI) was used to measure the efficiency of sampling from a posterior distribution of parameters in each iteration, which was computed by the following diagnostic [31]. \[\text{BFMI}\approx\frac{\sum\limits_{i=1}^{N}\left(E_{n}-E_{n-1}\right)^{2}}{ \sum\limits_{i=0}^{N}\left(E_{n}-\overline{E}\right)^{2}} \tag{28}\]  
In the HMC algorithm, the essence of BFMI computed by Equation (28) is the ratio of energy transition and marginal energy. If the energy transfer is too small, the sampling efficiency is low, indicating that the sampling parameters need to be adjusted. When BFMI approaches 1.0, it indicates that the sampling efficiency of parameters is high, and approximately independent samples can be collected from the posterior distribution. Figure 6 shows the BFMI analysis results of this case study. BFMI values of the two parallel Markov chains were 1.15 and 1.10, respectively. Moreover, it is evident that the energy transition almost shared the same distribution with the marginal energy, indicating that the sampling efficiency of the sampling algorithm was relatively high under the current parameter setting. To evaluate the efficacy of the improved Weibull model, we used the obtained sampling results to make a posterior prediction of the independent variables and compare them with the actual observed values. Figure 7 shows the comparison between the posterior prediction and the observed failure time of joint waterproof facilities, from which it can be seen that the posterior prediction results fitted well with the actual observation data. Meanwhile, it is evident that both the posterior prediction and the observed data showed an early failure tendency, which further demonstrates the validity of the model building. Figure 6: Analysis results of BFMI. Figure 8: Posterior distributions of the model parameters. Figure 10 shows the estimation results for joint waterproof facilities of the case study utility results in the accelerated wear-out stage where sharply increasing failure rates are expected after a relatively stable stage. The upper and lower bounds of the black line represent the warning time \(T_{0.8}\) (when the reliability drops to 80%) and the expected failure time \(E(T)\) respectively.