## 4 Bayesian Analysis of Weibull Mixture Distribution Using NG-RJMCMC Algorithm
### Determine the Appropriate Priors
#### 4.2.2 Priors and Posteriors  
In this section, we define the priors. In the hierarchical model in Equation (16), for each parameter, we assume that the priors are drawn independently. Based on research conducted by [PERSON] et al. [69], [PERSON] [70] and [PERSON] et al. [71], the priors for the location and scale parameters in the extreme value distribution are flat. In research by [PERSON] et al. [69], they chose the adoption of near-flat priors for the location and scale parameters. In research by [PERSON] and [PERSON] [70], the location and scale parameter priors are almost noninformative: the prior for \(\mu\) is extremely flat, while that for \(\sigma\) resembles \(\frac{1}{\sigma}\). Based on research by [PERSON] et al. [71], they have chosen a uniform distribution for the location and scale parameters. Therefore, in this study, the Gaussian distribution (with the large variance) with mean \(\varepsilon\) and variance \(\zeta^{2}\) was selected as a prior for location parameter \(\mu\). Thus, \(\mu_{j}\) for each component is given by  
\[f\left(\mu_{j}\big{|}\varepsilon,\zeta^{2}\right)=\frac{1}{\sqrt{2\pi\zeta^{2 }}}\exp\left\{-\frac{1}{2}\left(\frac{\mu_{j}-\varepsilon}{\zeta}\right)^{2} \right\}. \tag{18}\]  
Since the scale parameter \(\sigma\) controls the dispersion of the distribution, an appropriate prior is an inverse gamma distribution with the shape and scale parameters are \(\vartheta\) and \(\varpi\), respectively. This prior selection is supported by [PERSON] and [PERSON] [16], and [PERSON] [29]. Thus, \(\sigma_{j}\) for each component is given by  
\[f\big{(}\sigma_{j}|\vartheta,\varpi\big{)}=\frac{\varpi^{\vartheta}\exp\left( -\frac{\varpi}{\sigma_{j}}\right)}{\Gamma(\vartheta)\sigma_{j}^{\vartheta+1}}, \tag{19}\]  
where \(\varpi\sim\text{Gamma}(g,h)\). Used Equations (18) and (19), we get  
\[ \begin{array}{rcl}f(\theta|k,\xi)&=&\prod\limits_{j=1}^{k}f\left(\mu_{j}| \varepsilon,\zeta^{2}\right)f\big{(}\sigma_{j}|\vartheta,\varpi\big{)}\\ &=&\prod\limits_{j=1}^{k}\left[\frac{1}{\sqrt{2\pi\zeta^{2}}}\exp\left\{-\frac {1}{2}\left(\frac{\mu_{j}-\varepsilon}{\zeta}\right)^{2}\right\}\right] \left[\frac{\varpi^{\vartheta}\exp\left(-\frac{\varpi}{\sigma_{j}}\right)}{ \Gamma(\vartheta)\sigma_{j}^{\vartheta+1}}\right]\\ &=&\left(\frac{\varpi^{\vartheta}}{\Gamma(\vartheta)\sqrt{2\pi\zeta^{2}}} \right)^{k}\prod\limits_{j=1}^{k}\frac{\exp\left\{-\frac{1}{2}\left(\frac{\mu _{j}-\varepsilon}{\zeta}\right)^{2}\exp\left(-\frac{\varpi}{\sigma_{j}}\right) \right\}}{\sigma_{j}^{\vartheta+1}}\\ &=&\frac{\varpi^{\vartheta}}{\Gamma(\vartheta)^{2}(\pi_{j})^{\frac{1}{2}}\prod \limits_{j=1}^{k}\frac{\exp\left\{-\frac{1}{2}\left(\frac{\mu_{j}-\varepsilon} {\zeta}\right)^{2}-\frac{\varpi}{\sigma_{j}}\right\}}{\sigma_{j}^{\vartheta+1} }}\\ \end{array}  \tag{20}\]Therefore, the hyperparameter \(\zeta\) in Equation (17) is actually (\(\varepsilon\), \(\zeta^{2}\), \(\theta\), \(\omega\)). Thus, according to Equation (20) and the joint distribution in Equation (17), the fully conditional posterior distribution for \(\mu_{j}\) and \(\sigma_{j}\) are  
\[