## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
The analytically derived full conditional posterior hyperpriors (posterior hyperprior \(\propto\) prior \(\times\) hyperprior) utilized by the Gibbs sampler are shown below:  
\[\beta_{0}^{F}\mid\underline{m_{0}},\underline{\measuredangle},N,\mathbf{X}^{F}, \ \Sigma_{\beta_{0}}^{F}\sim\ \mathbf{MVN}\Bigg{(}\frac{\hat{\xi}}{\hat{\xi}+1} \underline{m_{0}}+\frac{1}{\hat{\xi}+1}\beta^{F},\ \frac{\Sigma_{\beta_{0}}^{F}}{\hat{\xi}+1}\Bigg{)}\] \[\Sigma_{\beta_{0}}^{F}\mid\underline{q_{0}},\underline{\measuredangle },N,\mathbf{X}^{F},\ \beta_{0}^{F}\sim\ \mathbf{MVN}\Big{(}\underline{q_{0}}+2,\ ( \beta_{0}^{F}-\beta^{F})(\beta_{0}^{F}-\beta^{F})^{T}\Big{)}\] \[+\hat{\varepsilon}(\beta_{0}^{F}-\underline{m_{0}})(\beta_{0}^{F }-\underline{m_{0}})^{T}+\Delta\Big{)}\] \[u_{0}^{F}\mid\underline{\measuredangle}_{u1^{\prime}},\underline{ \measuredangle}_{u2^{\prime}},N,\mathbf{X}^{F},v_{0}^{F}\propto\ \big{(}\psi_{j}-\frac{v_{0}^{F}}{2}\cdot \underline{\measuredangle}_{u1}\big{)}^{\big{(}u_{0}^{F}/2\big{)}-1}/\ \Gamma\Big{(}u_{0}^{F}/2\Big{)}^{\underline{\measuredangle}_{u2^{ \prime}}+1}\] \[v_{0}^{F}\mid\underline{\measuredangle}_{u1^{\prime}},\underline{ \measuredangle}_{u2^{\prime}},N,\mathbf{X}^{F},u_{0}^{F}\sim\ \mathbf{Ga}\Big{(}\underline{\measuredangle}_{v1}+\underline{u_{0}^{F}},\ \underline{\measuredangle}_{v2}+\psi_{j}\Big{)}\Bigg{\}}\text{for}\ \psi_{j} \tag{22}\]  
\[\beta_{0}^{S}\mid m_{0},\delta,Y,\mathbf{X}^{S},\ \Sigma_{\beta_{0}}^{S}\sim\ \mathbf{MVN}\Bigg{(}\frac{\delta}{\hat{\xi}+1}m_{0}+ \frac{1}{\hat{\xi}+1}\beta^{S},\ \frac{\Sigma_{\beta_{0}}^{S}}{\hat{\xi}+1}\Bigg{)}\] \[\Sigma_{\beta_{0}}^{S}\mid q_{0},\Lambda,Y,\mathbf{X}^{S},\ \beta_{0}^{S}\sim\ \mathbf{ IW}\Big{(}q_{0}+2,\ (\beta_{0}^{S}-\beta^{S})(\beta_{0}^{S}-\beta^{S})^{T}\Big{)}\] \[+\hat{\varepsilon}(\beta_{0}^{S}-m_{0})(\beta_{0}^{S}-m_{0})^{T}+ \Lambda\Big{)}\] \[u_{0}^{S}\mid\rho_{u1},\rho_{u2},Y,\mathbf{X}^{S},v_{0}^{S}\propto \ \big{(}\frac{1}{\sigma_{j}^{2}}\cdot\underline{\measuredangle}_{\rho 1}\big{)}^{\big{(}u_{0}^{S}/2\big{)}-1}/\ \Gamma\Big{(}u_{0}^{S}/2\Big{)}^{\rho_{u2}+1}\] \[\beta_{0}^{F}\mid\underline{m_{0}},\underline{\measuredangle},N,\mathbf{X}^{F}, \ \Sigma_{\beta_{0}}^{F}\sim\ \mathbf{MVN}\Bigg{(}\frac{\hat{\xi}}{\hat{\xi}+1} \underline{m_{0}}+\frac{1}{\hat{\xi}+1}\beta^{F},\ \frac{\Sigma_{\beta_{0}}^{F}}{\hat{\xi}+1}\Bigg{)}\]0  
Note that the expressions in Equations (21)-(24) are directly substituted into the joint conditional posterior of all parameters specified in Equation (20). Specifically, Equations (21)-(24) redefine the hyperprior terms \(p(\beta_{0}^{F}|\underline{m_{0}},\underline{\measuredangle})\)\(p(\Sigma_{\beta_{0}}^{F}|\underline{q_{0}},\underline{\measuredangle})\)\(p(\beta_{0}^{S}|m_{0},\delta)\)\(p(\Sigma_{\beta_{0}}^{S}|q_{0},\Lambda)\) and \(p(u_{0}^{F}|\underline{\measuredangle}_{u1^{\prime}},\underline{\measuredangle}_{u2})\)\(p(v_{0}^{F}|\underline{\measuredangle}_{u1^{\prime}},\underline{\measuredangle}_{u2})\)\(p(u_{0}^{S}|\rho_{u1},\rho_{u2})\)\(p(v_{0}^{S}|\rho_{v1},\rho_{v2})\) in Equation (20). For the main parameter term \(p(\beta_{j}^{F}|\beta_{0}^{F}\Sigma_{\beta_{0}}^{F})\)\(p(\beta_{j}^{S}|\beta_{0}^{S},\Sigma_{\beta_{0}}^{S})\)\(p(\psi_{j}|u_{0}^{F},v_{0}^{F})\)\(p(\sigma_{j}^{2}|u_{0}^{S},v_{0}^{S})\) in Equation (20); however, we employ the Metropolis-Hastings (MH) algorithm within the Gibbs sampler due to the lack of conjugate priors for the primary parameters \(\beta_{j}^{F},\psi_{j},\beta_{j}^{S},\sigma_{j}^{2}\) that align with our negative binomial and log-normal likelihoods. Given the two outcome models--claim counts \(N_{h}\) and claim amounts \(\bar{Y}_{h}\)--both Gibbs samplers can be executed in parallel to assess Equation (14a), assuming no covariate errors. Figure 2 illustrates the process of re-estimating the model parameters \(\beta_{j}^{F},\psi_{j},\beta_{j}^{S},\sigma_{j}^{2}\) for each cluster \(j\) using the two distinct Gibbs samplers.