## 3 Bayesian Coupled with Markov Chain Monte Carlo Approach
### Bayesian Approach for MSNBurr Mixture Model  
The prior distribution for MSNBurr was discussed by [PERSON] [16]. The proportion of the mixture components follows the Dirichlet distribution, as in the FSSN-MM. The following are prior distributions for each parameter of the MSNBurr mixture model (MSNBurr-MM):  
\[\mu_{j} \sim\text{Gaussian}\Big{(}\eta_{j},\varphi_{j}^{2}\Big{)},\] \[\sigma_{j} \sim\text{Invers Gamma}\Big{(}\alpha_{j},\beta_{j}\Big{)},\] \[\gamma_{j} \sim\text{Generalised Symmetrical Beta}\Big{(}\tau_{j},\tau_{j},l_{j},u_{j }\Big{)},\] \[p_{j} \sim\text{Dirichlet}(\delta_{1},\delta_{2},\ldots,\delta_{K}).\]  
The hyperparameters value are given as in the FSSN-MM; however, we set the hyperparameter for prior \(\gamma_{j}\) by \(\tau_{j}=\varphi_{j}/2\), \(l_{j}=0\), and \(u_{j}=2\). By setting these priors as independent priors and multiplying them with the likelihood, its joint posterior could be found easily (see Appendix A). The full conditional posterior distributions for each parameter are discussed below. For simplicity, the formulas are shown as logarithms (some derivations are shown in Appendix B) and all parts of the equation that do not contain the estimated parameter are assembled as a constant. The full conditional posterior for \(\mu_{j}\), \(j=1,2,\ldots,K\) is shown in Equation (12). \[ \begin{array}{ll}\ln f\Big{(}\mu_{j}|\sigma_{j},\gamma_{j},p_{j},\kappa_{j}, \mathbf{z},\mathbf{y}\Big{)}&=\text{constant}-\sum_{i=1}^{N}z_{ij}\bigg{(} \kappa\bigg{(}\frac{y_{iej}-\mu_{j}}{\sigma_{j}}\bigg{)}\bigg{)}-\frac{1}{2} \bigg{(}\frac{\mu_{j}-\eta_{j}}{\sigma_{j}}\bigg{)}^{2}\\ &+\sum_{i=1}^{N}z_{ij}\Bigg{(}-\big{(}\gamma_{j}+1\big{)}\ln\Bigg{(}1+\frac{ \exp\left(-\kappa\bigg{(}\frac{y_{iej}-\mu_{j}}{\sigma_{j}}\bigg{)}\bigg{)} \right)}{\gamma_{j}}\Bigg{)}\Bigg{)}\end{array}  \tag{14}\]  
The following Equation (15) shows the full conditional posterior for \(\sigma_{j}\), \(j=1,2,\ldots,K\):  
\[ \begin{array}{ll}\ln f\Big{(}\sigma_{j}|\mu_{j},\gamma_{j},p_{j},\kappa_{j},\mathbf{z},\mathbf{y}\Big{)}=\text{constant}-\Big{(}\alpha_{j}+1\Big{)}\ln \sigma_{j}-\frac{\beta_{j}}{\sigma_{j}}+\sum_{i=1}^{N}z_{ij}\ln\bigg{(}\frac{ \kappa_{j}}{\sigma_{j}}\bigg{)}+\\ \sum_{i=1}^{N}z_{ij}\Bigg{(}-\big{(}\gamma_{j}+1\big{)}\ln\Bigg{(}1+\frac{\exp \left(-\kappa\big{(}\frac{y_{iej}-\mu_{j}}{\sigma_{j}}\big{)}\right)}{\gamma_{ j}}\Bigg{)}\Bigg{)}+\sum_{i=1}^{N}z_{ij}\bigg{(}-\kappa_{j}\bigg{(}\frac{y_{iej}- \mu_{j}}{\sigma_{j}}\bigg{)}\bigg{)}\end{array}  \tag{15}\]  
The full conditional posterior of \(\gamma_{j}\), \(j=1,2,\ldots,K\) is given bellow:  
\[ \begin{array}{ll}\ln f\Big{(}\gamma_{j}|\mu_{j},\sigma_{j},p_{j},\kappa_{j},\mathbf{z},\mathbf{y}\Big{)}=\text{constant}-\Big{(}\tau_{j}-1\Big{)}\ln \Big{(}\big{(}\gamma_{j}-l_{j}\big{)}\Big{(}u_{j}-\gamma_{j}\big{)}\Big{)}+\\ \sum_{i=1}^{N}z_{ij}\Bigg{(}-\big{(}\gamma_{j}+1\big{)}\ln\Bigg{(}1+\frac{\exp \left(-\kappa\big{(}\frac{y_{iej}-\mu_{j}}{\sigma_{j}}\big{)}\right)}{\gamma_ {j}}\Bigg{)}\Bigg{)}+\sum_{i=1}^{N}z_{ij}\Big{(}-\kappa_{j}\bigg{(}\frac{y_{ iej}-\mu_{j}}{\sigma_{j}}\bigg{)}\Bigg{)}.\end{array}  \tag{16}\]  
The full conditional posterior for \(p_{j}\), \(j=1,\ 2,\ldots,\ K\) is as shown in Equation (12). The full conditional posterior of the label \(z_{ij}\) follows the multinomial \((1,\ \omega_{i1},\ldots,\ \omega_{iK})\) distribution, where \(\omega_{ij}\), \(i=1,\ 2,\ \ldots,\ N\), \(j=1,\ 2,\ldots,\ K\), as in Equation (13). The Gibbs sampling algorithm was constructed with a full conditional posterior for each parameter. The step of Gibbs sampling for the MSNBurr-MM is presented in Algorithm 2. ```
1:Initialize parameters \(\mu_{1}^{(0)},\ldots,\ \mu_{K}^{(0)},\sigma_{1}^{(0)},\ldots,\ \sigma_{K}^{(0)},\gamma_{1}^{(0)},\ldots,\ \gamma_{K}^{(0)},p_{1}^{(0)},\ldots,\ \rho_{K}^{(0)},\mathbf{z}_{1}^{(0)},\ldots,\ \mathbf{z}_{N}^{(0)}\), set \(t=1\), and determine \(T\). 2:While\(t\leq T\)do
3: Update value \(\mu_{j}\), \(j=1,\ 2,\ldots,\ K\) by generating \(\mu_{j}^{(t)}\) according to Equation(14).