## 3 Results and Discussion
### Goodness-of-Fit of the Distribution  
In this subsection, the authors analyze the goodness-of-fit of the distribution to the truth data; maximum likelihood estimation (MLE) was used to evaluate the accuracies of the corresponding distributions obtained using hyperparameters estimated using BMCM, ESM, and MMR. MLE is a method to evaluate models using specific data--given different prior to distributions; it determines the corresponding likelihoods of the actual sample to determine the confidences of the prior distributions. The likelihood is termed the confidence factor; and it is proportional to the proximity of the prior distribution to the ground truth. Obviously, the distribution that is closest to the ground truth is deemed to be the best. MLE was conducted using the following procedure. Let us assume that the failure probability distribution of the web service composition, \(X=\left\{p_{1},p_{2}\ldots p_{n}\right\}\), is given by \(f(X\big{|}\theta)\), where \(\theta\) denotes the hyperparameters, \(a\) and \(b\), and the density function of the prior distribution of \(\theta\) is \(\pi(\theta)\). After obtaining \(X\), the density function \(\pi(\theta\big{|}X)\) of the posterior distribution can be expressed as  
\[\pi(\theta|X)=\frac{f(X\big{|}\theta)\pi(\theta)}{\int_{\theta}f(X\big{|}\theta )\pi(\theta)d_{\theta}} \tag{23}\]  
where \(f(X\big{|}\theta)\) reflects the knowledge of \(\theta\) based on the observed \(X\). If there are two parameters, \(\theta_{1}\) and \(\theta_{2}\), the authors obtain  
\[f(X\big{|}\theta_{1})>f(X\big{|}\theta_{2}) \tag{24}\]  
From the perspective of statistics, the probability of observed \(X\) with the parameter \(\theta_{1}\) is higher than \(X\) with \(\theta_{2}\). Thus, (24) can be interpreted as the likelihood of \(\theta=\theta_{1}\) being larger than that of \(\theta=\theta_{2}\). Thus, when \(X\) is constant, \(f(X\big{|}\theta)\) is independent of the prior distribution of \(\theta\) as the function with \(\theta\); instead, it merely reflects the characteristics of the sample. The denominator of (23) is given by  
\[m(X)=\int_{\theta}f(X|\theta)\pi(\theta)d_{\theta} \tag{25}\]  
This is the marginal distribution of \(X\) that depends on \(f(X\big{|}\theta)\). \(f(X\big{|}\theta)\) is the conditional distribution density and, for a given \(\theta\), the marginal distribution describes the action of the sample \(X\) under the prior distribution, \(\pi(\theta)\), and the sample follows this distribution. If \(\pi(\theta)\) follows a certain distribution, the parameter of \(\pi(\theta)\) is obtained from the maximum value of \(m(X)\)--i.e., the MLE. If there are \(m\) prior distributions that follow \(f(X\big{|}\theta)\), the failure probability samples are \(p_{1},p_{2}\ldots p_{n}\). Let \(\pi_{i}(\theta)\) be the \(i^{\text{th}}\) prior distribution based on the sample. If the ground truth prior distribution is denoted by \(\pi(\theta)\), the \(m\) prior distributions can be integrated into the ground truth prior distribution using the following equation. \[\pi(\theta)=\sum\
olimits_{i=1}^{m}\varepsilon_{i}\pi_{i}(\theta) \tag{26}\]  
where \(\varepsilon_{i}\) denotes the weight of \(i\)th prior distribution, \(\pi_{i}(\theta)\), and \(\sum_{i=1}^{m}\varepsilon_{i}=1\). The marginal distribution of the prior distribution, \(\pi_{i}(\theta)\), is given by  
\[m(x|\pi_{i})=\int_{\theta}f(X|\theta)\pi_{i}(\theta)d_{\theta} \tag{27}\]  
The marginal distribution of the ground truth prior distribution, \(\pi(\theta)\), is given by  
\[m(x|\pi)=\int_{\theta}f(X\Big{|}\theta)\pi(\theta)d_{\theta} \tag{28}\]  
The failure data, \(p_{1},p_{2}\ldots p_{n}\), generated by the marginal distribution, \(m(x|\pi_{i})\), can be used to obtain the following likelihood function. \[L(X|\pi_{i})=\prod_{i=1}^{n}m(x_{i}|\pi_{i}) \tag{29}\]  
By MLE, \(L(X|\pi_{i})\) is bigger to represent the closer proximity to the ground truth, the weight of \(\pi_{i}(\theta)\) is bigger, high probability to choose this prior distribution. The confidence factor can be obtained as follows.