## 4 Bayesian Analysis of Weibull Mixture Distribution Using NG-RJMCMC Algorithm
### RJMCMC Move Types for EV-I Mixture Distribution
#### 4.3.1 Split and Combine Moves  
For move (e), we choose between split or combine, with the probabilities \(b_{k}\) and \(d_{k}=1-b_{k}\), respectively, depending on \(k\). Note that \(d_{1}=0\) and \(b_{k_{\max}}=0\), where \(k_{\max}\) is the maximum value for \(k\); otherwise, we choose \(b_{k}=d_{k}=0.5\), for \(k=1,2,\cdots,k_{\max}-1\). The combining proposal works as follows: choose two components \(j_{1}\) and \(j_{2}\), where \(\mu_{1}<\mu_{2}\) with no other \(\mu_{j}\in[\mu_{1},\mu_{2}]\). If these components are combined, we reduce \(k\) by \(1\), which forms a new component \(j^{*}\) containing all the observation previously allocated to \(j_{1}\) and \(j_{2}\), and then creates values for \(w_{j^{*}}\), \(\mu_{j^{*}}\), and \(\sigma_{j^{*}}\) by preserving the first two moments, as follows:  
\[\left. \begin{array}{c}w_{j^{*}}=w_{j_{1}}+w_{j_{2}}\\ w_{j^{*}}\mu_{j^{*}}=w_{j_{1}}h_{j_{1}}+w_{j_{2}}\mu_{j_{2}}\\ w_{j^{*}}\left(\mu_{j^{*}}^{2}+\sigma_{j^{*}}\right)=w_{j_{1}}\left(\mu_{j_{1 }}^{2}+\sigma_{j_{1}}\right)+w_{j_{2}}\left(\mu_{j_{2}}^{2}+\sigma_{j_{2}} \right).\end{array} \right\} \tag{28}\]  
Figure 1: Graphical representation of the Bayesian hierarchical finite EV-I mixture distribution. Nodes in this graph represent random variables, green boxes are fixed hyperparameters, boxes indicate repetition (with the number of repetitions in the upper left), arrows describe conditional dependencies between variables, and the red arrow describes a variable transformation. The splitting proposal works as follows: a random \(j^{*}\) component is selected then split into two new components, \(j_{1}\) and \(j_{2}\), with the weights and parameters (\(w_{j_{1}}\), \(\mu_{j_{1}}\), \(\sigma_{j_{1}}\)) and (\(w_{j_{2}}\), \(\mu_{j_{2}}\), \(\sigma_{j_{2}}\)), respectively, conforming to Equation (28). Based on this information, we have three degrees of freedom, so we generate three random numbers \(\mathbf{u}=(u_{1},u_{2},u_{3})\), where \(u_{1}\sim\text{Beta}(2,2)\), \(u_{2}\sim\text{Beta}(2,2)\), and \(u_{3}\sim\text{Beta}(1,1)\)[16]. Then, split transformations are defined as follows:  
\[ \begin{array}{c}w_{j_{1}}=w_{j^{*}}u_{1},w_{j_{2}}=w_{j^{*}}(1-u_{1})\\ \mu_{j_{1}}=\mu_{j^{*}}-u_{2}\sqrt{\sigma_{j^{*}}}\sqrt{\frac{w_{j_{2}}}{w_{j_ {1}}}},\mu_{j_{2}}=\mu_{j^{*}}+u_{2}\sqrt{\sigma_{j^{*}}}\sqrt{\frac{w_{j_{1}} }{w_{j_{2}}}}\\ \sigma_{j_{1}}=u_{3}\big{(}1-u_{2}^{2}\big{)}\sigma_{j^{*}}\frac{w_{j^{*}}}{w_ {j_{1}}},\sigma_{j_{2}}=(1-u_{3})\big{(}1-u_{2}^{2}\big{)}\sigma_{j^{*}}\frac{ w_{j^{*}}}{w_{j_{2}}}.\end{array}  \tag{29}\]  
Then, we compute the acceptance probabilities of split and combine moves: \(\min\{1,A\}\) and \(\min\big{\{}1,A^{-1}\big{\}}\), respectively. According to Equation (9), we obtain \(A\) as in Equation (30):  
\[ \begin{array}{c}A\quad=\frac{\Pr(z,w;k+1,\theta,\zeta,\zeta,\theta,\omega|y )d_{k+1}}{\Pr(z,w,k,\theta,\zeta,\theta,\omega|y)b_{1}\mu_{\text{side}}\sigma (\mathbf{u})}\Big{|}\frac{\partial\Delta^{*}}{\partial(\Delta,\mathbf{u})} \Big{|}\\ \quad=\text{likelihood ratio}\left[\frac{f(k+1)}{f(k)}\right](k+1)\frac{w_{j_{1}} ^{\delta+l_{1}-1}w_{j_{2}}^{\delta+l_{2}-1}}{w_{j_{2}}^{\delta+l_{1}-2}B( \delta,\delta)}\\ \quad\times\left(\frac{1}{2\pi\ell^{2}}\right)^{\frac{1}{2}}e^{-\frac{1}{2 \ell^{2}}\left(w_{j_{1}}-\epsilon\right)^{2}+\left(w_{j_{2}}-\epsilon\right)^ {2}-\left(w_{j_{2}}-\epsilon\right)^{2}}\\ \quad\times\frac{\sigma^{\theta}}{\Gamma(\theta)}\left(\frac{\sigma_{j^{*}}}{ \sigma_{j_{1}}}\right)^{\theta+1}\exp\left(-\sigma\Big{(}\frac{1}{\sigma_{j_{ 1}}}+\frac{1}{\sigma_{j_{2}}}-\frac{1}{\sigma_{j^{*}}}\Big{)}\right)\\ \quad\times\frac{d_{k+1}}{b_{k}P_{\text{side}}\delta_{22}(u_{1})\delta_{22}(u _{1})\delta_{11}(u_{3})}\\ \quad\times\frac{w_{j_{1}}\left|w_{j_{1}}-\mu_{j_{2}}P_{j_{1}}\tau_{2}}{u_{2} \big{(}1-u_{2}\big{)}u_{3}(1-u_{3})\sigma_{j^{*}}}\end{array}