## Appendix A Joint Posterior Density for Nenomimo  
Assuming that each parameter in the Nenomimo, \(\mathbf{\upmu}\), \(\mathbf{\upsigma}\), \(\mathbf{\upgamma}\), and \(\mathbf{\uprho}\), is independent (where \(\mathbf{\upmu}=\left(\mu_{1},\mu_{2},\ldots,\mu_{j},\ldots,\mu_{k}\right)\), \(\mathbf{\upsigma}=\left(\sigma_{1},\sigma_{2},\ldots,\sigma_{j},\ldots,\sigma_{ k}\right)\), then \(\mathbf{\upgamma}=\left(y_{1},y_{2},\ldots,y_{i},\ldots,y_{N}\right)\) is the pixels in the MRI image. Therefore, the joint prior density for each \(\mathbf{\upmu}\), \(\mathbf{\upsigma}\), and \(\mathbf{\upgamma}\) is given by:  
\[g(\mathbf{\upmu}) =g(\mu_{1})\times g(\mu_{2})\times\ldots\times g(\mu_{K})=\prod _{j=1}^{K}g(\mu_{j}),\] \[g(\mathbf{\upsigma}) =g(\sigma_{1})\times g(\sigma_{2})\times\ldots\times g(\sigma_{ K})=\prod_{j=1}^{K}g(\sigma_{j}),\] \[g(\mathbf{\upgamma}) =g(\gamma_{1})\times g(\gamma_{2})\times\ldots\times g(\gamma_{ K})=\prod_{j=1}^{K}g(\gamma_{j}).\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Sample** **Image**} & \multirow{2}{*}{\(j\)} & \multicolumn{4}{c}{**FSSN-MM**} & \multicolumn{4}{c}{**MSNBurr-MM**} \\ \cline{3-10}  & & & \(p_{j}\) & \(\mu_{j}\) & \(\sigma_{j}\) & \(\gamma_{j}\) & \(p_{j}\) & \(\mu_{j}\) & \(\sigma_{j}\) & \(\sigma_{j}\) \\ \hline \multirow{3}{*}{**IM01**} & 1 & 0.084 & 1.569 & 4.856 & 26.468 & 0.076 & 1.575 & 3.774 & 14.024 \\  & 2 & 0.862 & 89.569 & 7.992 & 0.904 & 0.861 & 87.891 & 6.678 & 0.864 \\  & 3 & 0.054 & 165.001 & 6.719 & 3.566 & 0.063 & 163.257 & 7.138 & 2.555 \\ \hline \multirow{3}{*}{**IM02**} & 1 & 0.163 & 2.684 & 4.777 & 4.625 & 0.186 & 2.887 & 4.576 & 4.128 \\  & 2 & 0.487 & 113.521 & 7.562 & 11.061 & 0.501 & 112.372 & 5.123 & 10.784 \\  & 3 & 0.196 & 158.01 & 5.324 & 7.225 & 0.196 & 154.753 & 4.852 & 8.211 \\  & 4 & 0.154 & 197.731 & 7.789 & 0.916 & 0.117 & 198.262 & 5.476 & 0.913 \\ \hline \multirow{3}{*}{**IM03**} & 1 & 0.079 & 0.379 & 2.149 & 10.251 & 0.074 & 0.375 & 2.14 & 9.321 \\  & 2 & 0.813 & 103.168 & 6.611 & 1.083 & 0.839 & 103.194 & 6.62 & 0.992 \\  & 3 & 0.108 & 197.113 & 7.708 & 13.934 & 0.087 & 197.263 & 7.685 & 18.858 \\ \hline \multirow{3}{*}{**IM04**} & 1 & 0.054 & 0.475 & 2.379 & 17.343 & 0.069 & 0.445 & 3.469 & 16.321 \\  & 2 & 0.665 & 96.015 & 5.555 & 7.236 & 0.564 & 122.981 & 6.278 & 5.048 \\ \cline{1-1}  & 3 & 0.136 & 125.176 & 5.355 & 6.725 & 0.245 & 161.062 & 3.283 & 4.769 \\ \cline{1-1}  & 4 & 0.145 & 200.36 & 7.18 & 12.089 & 0.122 & 199.908 & 4.476 & 8.866 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameter of the Nenomimo for the sample image. Via the Bayes theorem, the joint posterior density of the Nenomimo is:  
\[f\big{(}\mu,\sigma,\gamma,\mathbf{p},\mathbf{z}|\mathbf{y}\big{)}=\frac{f( \mathbf{y},\mu,\sigma,\gamma,\mathbf{p},\mathbf{z})}{f(\mathbf{y})}\propto f( \mathbf{y},\mu,\sigma,\gamma,\mathbf{p},\mathbf{z}).\]  
Substituting the joint density for each prior in MSNBurr-MM, we then obtain:  
\[ \begin{array}{l}f(\mathbf{y},\mu,\sigma,\gamma,\mathbf{p},\mathbf{z})=f( \mathbf{y},\mathbf{z}|\mu,\sigma,\gamma,\mathbf{p})g(\mu)g(\sigma)g(\gamma)g( \mathbf{p})=\\ \frac{\prod_{i=1}^{N}\prod_{j=1}^{K}\left(p_{j}\frac{\kappa}{\sigma}\exp\left(- \kappa\left(\frac{y-\mu}{\sigma}\right)\right)\right)(1+\\ \frac{\exp\left(-\kappa\left(\frac{r-\mu}{\sigma}\right)\right)}{\gamma}\right) ^{-(\gamma+1)}\gamma\prod_{j=1}^{K}\left(\mu_{j}\right)g\big{(}\sigma_{j} \big{)}g\big{(}\gamma_{j}\big{)}\times(K-1)!.\end{array}  \tag{10}\]  
Since:  
\[ \begin{array}{l}\mu_{j}\sim\text{Gaussian}\Big{(}\eta_{j},\varphi_{j}^{2} \Big{)},\\ \sigma_{j}\sim\text{Invers Gamma}\big{(}\alpha_{j},\beta_{j}\big{)},\\ \gamma_{j}\sim\text{Generalised Symmetrical Beta}\Big{(}\tau_{j},\tau_{j},l_{j},u_{j} \Big{)},\\ p_{j}\sim\text{Dirichlet}(\delta_{1},\delta_{2},\dots,\delta_{K}),\end{array} \]  
We then get the following in detail. 1.