## 3 Results and Discussion
### Goodness-of-Fit of the Distribution
\[\varepsilon_{k}=\frac{L(X|\pi_{i})}{\sum_{i=1}^{i}L(X|\pi_{i})} \tag{30}\]  
The confidence factor represents the probability of a prior distribution being close to the truth distribution. Thus, if the confidence factor of a distribution is high, it is highly feasible to choose it as a prior distribution. Next, the authors compared the accuracies of the distribution constructed using the hyperparameters estimated using BMCM, ESM, and MMR. The failure probability follows \(Beta(a,b)\), where \(a\) and \(b\) are hyperparameters. First, certain defects were introduced into the web services, following the discussion presented in [32]. The web service composition considered in this section is depicted in Figure 4. The actual failure probability following the introduction of the defects was regarded as the ground truth, the failure probabilities obtained via various method were regarded as evaluation values, \(X\) denotes the ground truth data, and MLE was used to determine the distribution that is closest to the truth. The failure probabilities of each web service obtained via module-level testing are presented in Table 5--\(\hat{p}_{1}=0.011,\hat{p}_{2}=0.034\), \(\hat{p}_{3}=0.096\) and \(\hat{p}_{4}=0.018\). Based on injected defects, the ground truth of the failure probability of the web service composition was calculated to be \(X=(0.184,\ 0.299,\ 0.156)\), which follows \(Beta(3.0,\ 6.0)\). Then, the authors calculated the failure probabilities using the predictions obtained from BMCM, ESM, and MMR as evaluation values. In the case of BMCM, the initial non-informative prior distribution was assumed to be \(Beta(2,\ 2)\). In the case of ESM, after assessment over 10 h, the prior distribution of the web service composition was estimated to by expert interview and the initial prior distribution was \(Beta(3,\ 6.3)\). In the case of MMR, by (20) and (22), the initial prior distribution was \(Beta(2.5,\ 3.9)\). After three steps of growth assessment, the probability distributions corresponding to the three methods were observed to be \(\pi_{1}=Beta(3.1,\ 5.9)\), \(\pi_{2}=Beta(2.7,\ 6.6)\), and \(\pi_{3}=Beta(3.5,\ 7)\) (\(\pi_{1}\) denotes MMR, \(\pi_{2}\) denotes BMCM, and \(\pi_{3}\) denotes ESM). \begin{table}
\begin{tabular}{c c} \hline \hline
**Web Services** & **Failure Probability** \\ \hline \(w_{1}\) & 0.011 \\ \(w_{2}\) & 0.034 \\ \(w_{3}\) & 0.096 \\ \(w_{4}\) & 0.018 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Failure probabilities of web services. By (27), the marginal distribution functions are given by  
\[ \begin{array}{l}m(x|\pi_{1})=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x )^{b-1}=\frac{\Gamma(9)}{\Gamma(3.1)\Gamma(5.9)}x^{2.1}(1-x)^{4.9}\\ m(x|\pi_{2})=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}=\frac{ \Gamma(9.3)}{\Gamma(2.2)\Gamma(6.6)}x^{1.7}(1-x)^{5.6}\\ m(x|\pi_{3})=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}=\frac{ \Gamma(10.5)}{\Gamma(3.5)\Gamma(7)}x^{2.5}(1-x)^{6}\end{array} \]  
Then, by (29), the likelihood functions are given by  
\[ \begin{array}{l}L(X|\pi_{1})=8.3003e-005\\ L(X|\pi_{2})=4.0761e-005\\ L(X|\pi_{3})=1.2588e-005\end{array} \]  
By (30), the respective confidences are given by  
\[ \begin{array}{l}\varepsilon_{1}=0.6707\\ \varepsilon_{2}=0.3294\\ \varepsilon_{3}=0.1321\end{array} \]  
As to \(\varepsilon_{1}>\varepsilon_{2}>\varepsilon_{3}\), the prior distribution corresponding to MMR is closest to the ground truth and is, therefore, the best candidate for the prior distribution. The curves of \(Beta\) corresponding to the three different methods are illustrated in Figure 6. Line 0 corresponds to the ground truth, Line 1 corresponds to the prior distribution, \(\pi_{1}=Beta(3.1,\ 5.9)\), Line 2 corresponds to the prior distribution, \(\pi_{2}=Beta(2.7,\ 6.6)\), and Line 3 corresponds to the prior distribution, \(\pi_{3}=Beta(3.5,\ 7)\). Line 1, corresponding to MMR, is evidently a significantly better fit compared to the other curves. Therefore, calculating the hyperparameters using MMR is the most reasonable approach in this case.