## 3 Bayesian Coupled with Markov Chain Monte Carlo Approach
### Bayesian Approach for FSSN Mixture Model  
The Bayesian approach was employed to perform the estimation, owing to the non-close form of the posterior distribution that acts as the multiplication between the likelihood function in Equation (8) and the prior distribution. We designed the prior distribution for the FSSN mixture model (FSSN-MM) following [PERSON] et al. (2017). The prior distribution of mixture parameters were discussed by [PERSON] et al. (2022). \[ \begin{array}{ll}\mu_{j}&\sim\text{Gaussian}\Big{(}\eta_{j},\varphi_{j}^{2} \Big{)},\\ \sigma_{j}&\sim\text{Invers}\;\text{Gamma}\big{(}\alpha_{j},\beta_{j}\big{)}, \\ \gamma_{j}&\sim\text{Gamma}\big{(}a_{j},b_{j}\big{)},\\ p_{j}&\sim\text{Dirichlet}(\delta_{1},\delta_{2},\ldots,\delta_{K}).\end{array} \]  
The value of hyperparameters were calculated from the data. The hyperparameter \(\eta_{j}=\text{mean}\left(y_{i\in j}\right)\) and \(\varphi_{j}=\text{var}\big{(}y_{i\in j}\big{)}\), where \(y_{i\in j}\) represents all data that belongs to the \(j\)th cluster. We also set \(\alpha_{j}=\varphi_{j}\), \(\beta_{j}=\eta_{j}/\left(\alpha_{j}-1\right)\), \(a_{j}=\eta_{j}^{2}/\varphi_{j}\), and \(b_{j}=\eta_{j}/\varphi_{j}\). By employing these priors as independent priors and contaminating them with the likelihood, the joint posterior can be constructed easily by multiplying them together. This derivation and the Gibbs sampling algorithm for the FSSN mixture model were adopted from [PERSON] et al. (2017). The full conditional posterior was conducted to generate the parameter values. It was derived from multiplying the likelihood and the prior distribution of parameters, which were estimated ([PERSON], 2020). The full conditional posterior of each parameter is given below (for simplicity, all equations are written in logarithm and all parts of equation that do not contain the estimated parameter are summed as a constant). (a) The full conditional posterior for the location parameter \(\mu_{j}\), \(j=1,2,\ldots,K\) is given by:  
\[ \begin{array}{ll}\ln f\big{(}\mu_{j}|\sigma_{j},\gamma_{j},p_{j},\mathbf{z}, \mathbf{y}\big{)}&=\text{constant}-\Big{(}\alpha_{j}+1\Big{)}\ln\!\big{(} \sigma_{j}\big{)}+\frac{\beta_{j}}{\sigma_{j}}\\ &+-n\log\!\left(\sigma_{j}\gamma_{j}\sqrt{2\pi}+\frac{\sigma_{j}\sqrt{2\pi}}{ \gamma_{j}}\right)-\frac{1}{2}\sum_{i=1}^{N}z_{ij}\!\left(\frac{y_{i\in j}- \mu_{j}}{\sigma_{j}}\right)^{2}.\end{array}  \tag{9}\]  
(b) The full conditional posterior for parameter \(\sigma_{j}\), \(j=1,\ 2,\ \ldots,\ K\) is given by Equation (10):  
\[ \begin{array}{ll}\ln f\Big{(}\gamma_{j}|\mu_{j},\sigma_{j},p_{j},\mathbf{z}, \mathbf{y}\Big{)}&=\text{constant}-\Big{(}\alpha_{j}+1\Big{)}\ln\!\big{(} \gamma_{j}\big{)}-b_{j}\gamma_{j}\\ &+\sum_{i=1}^{N}z_{ij}\log\frac{1}{\gamma^{2}}I_{[\mu,255)}\big{(}y_{i\in j} \big{)}+\gamma^{2}I_{(0,\mu)}\big{(}y_{i\in j}\big{)}\\ &-n\log\!\left(\sigma_{j}\gamma_{j}\sqrt{2\pi}+\frac{\sigma_{j}\sqrt{2\pi}}{ \gamma_{j}}\right)\end{array}  \tag{10}\]  
(c) Equation (11) shows the full conditional posterior for parameter \(\gamma_{j}\), \(j=1,\ 2,\ \ldots,\ K\):  
\[ \begin{array}{ll}\ln f\Big{(}\gamma_{j}|\mu_{j},\sigma_{j},p_{j},\mathbf{z}, \mathbf{y}\Big{)}&=\text{constant}+\Big{(}a_{j}-1\Big{)}\ln\!\big{(}\gamma_{j} \big{)}-b_{j}\gamma_{j}\\ &+\sum_{i=1}^{N}z_{ij}\log\frac{1}{\gamma^{2}}I_{[\mu,255)}\big{(}y_{i\in j} \big{)}+\gamma^{2}I_{(0,\mu)}\big{(}y_{i\in j}\big{)}\\ &-n\log\!\left(\sigma_{j}\gamma_{j}\sqrt{2\pi}+\frac{\sigma_{j}\sqrt{2\pi}}{ \gamma_{j}}\right)\end{array}  \tag{11}\]
4. Equation (12) shows the full conditional posterior distribution parameter \(p_{j}\), \(j=1,\;2,\;\ldots,\;K\): \[f\Big{(}p_{j}\big{|}\sigma_{j},\gamma_{j};\mu_{j};\mathbf{z},\mathbf{y}\Big{)} \propto\prod_{j=1}^{K}p_{j}^{\sum_{i=1}^{N}z_{ij}}=\prod_{j=1}^{K}p_{j}^{(\sum _{i=1}^{N}z_{ij}+1)-1}.\] (12) This is a Dirichlet \(\Big{(}1+\sum_{i=1}^{N}z_{i1},\ldots,1+\sum_{i=1}^{N}z_{iK}\Big{)}\) distribution. 5. The label \(z_{ij}\), \(i=1,2,\ldots,N\) and \(j=1,2,\ldots,K\), only has two possible values, i.e., 0 or 1.