## 4 Bayesian Method
### Posterior Analysis  
To apply the importance sampling method, posterior density function is first expressed as shown below:  
\[\pi(\alpha,\lambda,\beta\mid x)\propto g_{\lambda}(u_{1},v_{1})\times g_{ \alpha|\lambda}(u_{2},v_{2})\times g_{(\beta-1)|\alpha,\lambda}(u_{3},v_{3}) \times q(\alpha,\lambda,\beta), \tag{14}\]  
where \(g_{\lambda}(u_{1},v_{1})\), \(g_{\alpha|\lambda}(u_{2},v_{2})\), and \(g_{(\beta-1)|\alpha,\lambda}(u_{3},v_{3})\) represent gamma density functions with parameters given by:  
\[ \begin{split} u_{1}=&\ m_{1}+m_{2}-r_{1}-r_{2}+a_{2 }\\ v_{1}=&\ b_{2}+\sum_{i=r_{1}+1}^{m_{1}}\frac{1}{x_{ 1i}}+\sum_{i=r_{2}+1}^{m_{2}}\frac{1}{x_{2i}},\\ u_{2}=&\ m_{1}+m_{2}-r_{1}-r_{2}+a_{1}\\ v_{2}=&\ b_{1}-\sum_{i=r_{1}+1}^{m_{1}}(1+R_{1i}) \ln(1-e^{-\lambda/x_{1i}})-\sum_{i=r_{2}+1}^{m_{2}}(1+R_{2i})\ln(1-e^{-\lambda /x_{2i}})\end{split} \]  
and  
\[ \begin{split} u_{3}=&\ \frac{1}{2}m_{2}+\frac{1}{2}r_{2}+a_{3 }\\ v_{3}=&\ b_{3}-\sum_{i=r_{2}+1}^{m_{2}}\alpha(1+R_{ 2i})\ln(1-e^{-\lambda/x_{2i}}),\end{split} \]  
also, \(q(\alpha,\lambda,\beta)\) is given by:  
\[q(\alpha,\lambda,\beta)=\frac{q^{\prime}(\alpha,\lambda,\beta)}{v_{2}^{u_{2} }v_{3}^{u_{3}}}, \tag{15}\]where  
\[ \begin{split} q^{\prime}(\alpha,\lambda,\beta)=&\beta^{m_{2 }-r_{2}}(\beta-1)^{\frac{m_{1}}{2}-\frac{m_{2}}{2}}\Big{[}1-(1-e^{-\lambda/x_{1 _{D}+1}})^{a}\Big{]}^{r_{1}}\Big{[}1-(1-e^{-\lambda/x_{2_{D}+1}})^{\beta a} \Big{]}^{r_{2}}\\ &\times\exp\Bigg{\{}-\sum_{i=r_{1}+1}^{m_{1}}\ln\Big{[}x_{1i}^{2 }(1-e^{-\lambda/x_{1i}})\Big{]}-\sum_{i=r_{2}+1}^{m_{2}}\ln\Big{[}x_{2i}^{2}(1- e^{-\lambda/x_{2i}})\Big{]}\Bigg{\}}.\end{split}  \tag{16}\]  
After that, it is possible to utilize the importance sampling to derive the Bayesian estimation and the relative BCIs. The specific steps of the method are shown below:  
Step 1: Generate a random variable \(\lambda_{1}\) from \(\text{Gamma}(u_{1},v_{1})\). Step 2: Generate a random variable \(\alpha_{1}\) from \(\text{Gamma}(u_{2},v_{2})\). Step 3: Generate a random variable \(\beta_{1}^{{}^{\prime}}\) from \(\text{Gamma}(u_{3},v_{3})\), and set \(\beta_{1}=\beta_{1}^{{}^{\prime}}+1\). Step 4: Perform Steps 1-3 a total of \(D\) times to generate \((\lambda_{1},\cdots,\lambda_{D})\), \((\alpha_{1},\cdots,\alpha_{D})\) and \((\beta_{1},\cdots,\beta_{D})\). Step 5: Derive the BEs under SELF as  
\[\bar{\alpha}=\frac{\sum_{i=1}^{D}\alpha_{i}q(\alpha_{i},\lambda_{i},\beta_{i} )}{\sum_{i=1}^{D}q(\alpha_{i},\lambda_{i},\beta_{i})},\quad\bar{\lambda}=\frac {\sum_{i=1}^{D}\lambda_{i}q(\alpha_{i},\lambda_{i},\beta_{i})}{\sum_{i=1}^{D}q (\alpha_{i},\lambda_{i},\beta_{i})}\]  
and  
\[\bar{\beta}=\frac{\sum_{i=1}^{D}\beta_{i}q(\alpha_{i},\lambda_{i},\beta_{i})}{ \sum_{i=1}^{D}q(\alpha_{i},\lambda_{i},\beta_{i})}.\]  
The \(100(1-\tau)\%\) HPDCIs are constructed through the same method of [22]. For the sake of concision, only the BCI and corresponding HPDCI of the parameter \(\beta\) are discussed here, the intervals for the other two parameters can be obtained using the same procedure. Assume that for \(0<p<1\), \(\beta_{p}\) satisfies \(P[\beta\leq\beta_{p}\mid x]=p\). Then a more intricate approach involves deriving a consistent estimator of \(\beta_{p}\) using the SELF based on the produced sample \((\alpha_{1},\lambda_{1},\beta_{1}),\cdots,(\alpha_{D},\lambda_{D},\beta_{D})\). Let  
\[u_{i}=\frac{q(\alpha_{i},\lambda_{i},\beta_{i})}{\sum_{i=1}^{D}q(\alpha_{i}, \lambda_{i},\beta_{i})},\quad i=1,\cdots D,\]  
reorganize \(\{(\beta_{1},u_{1}),\cdots,(\beta_{D},u_{D})\}\) to be \(\{(\beta_{(1)},u_{(1)}),\cdots,(\beta_{(D)},u_{(D)})\}\), in which \(\beta_{(1)}<\cdots<\beta_{(D)}\). It is worth noting that the \(u_{(i)}\)'s are not arranged in any specific order, they are simply correlated with the \(\beta_{(i)}\)'s.