## 3 Modeling Method for \(\mathbf{S_{h}}|\mathbf{X}^{F},\mathbf{X}^{S}\)
### Introduction of Gustafson Correction to Address NDB Covariate: RQ2
For a concise comparison, refer to Table 1.  
The implication is that when we align the two models by integrating out the unobservable true covariate \(\mathbf{x}\) from the complete joint model, the resulting equation reveals the connection between the parameters of the complete joint model and those of the incomplete joint model as follows:  
\[\int f(Y,\mathbf{x}^{*},\mathbf{x}\mid\mathbf{z})\ d\mathbf{x} = f(Y,\mathbf{x}^{*}\mid\mathbf{z}) \tag{10}\]  
While explicitly solving the integral in Equation (10) can be difficult, comparing the parameterizations on both sides of the equation uncovers the link between the parameters of the model based on the NDB covariate \(\mathbf{x}^{*}\) and the true model using the actual covariate \(\mathbf{x}\). Motivated by [PERSON] (2008) [15] and [PERSON] et al. (2021) [8], we analytically derive the solution to the integral in Equation (10), with the detailed derivation provided in Part 3 of the Supplementary Materials. Accordingly, a hybrid Gibbs sampler is developed using the relationship embedded in the parameterizations (highlighted in Part 2 of the Supplementary Materials) to mitigate the model risk associated with the NDB covariates. \begin{table}
\begin{tabular}{c c} \hline \hline
**Complete Joint (Unknown)** & **Incomplete Joint (Known)** \\ \hline \(\frac{f(Y|\mathbf{x},\mathbf{z})}{\text{outcome}}\cdot\frac{f(\mathbf{x}^{*}| \mathbf{x})}{\text{measurement}}\cdot\frac{f(\mathbf{x}|\mathbf{z})}{\text{ exposure}}\) & \(\frac{f(Y|\mathbf{x}^{*},\mathbf{z})}{\text{outcome}}\cdot\frac{f(\mathbf{x}^{*}| \mathbf{z})}{\text{exposure}}\) \\ \hline \(=f(Y,\mathbf{x}^{*},\mathbf{x}\mid\mathbf{z})\) & \(=f(Y,\mathbf{x}^{*}\mid\mathbf{z})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of complete and incomplete joint models. Clustering \(S_{h}|\mathbf{X}^{F},\mathbf{X}^{5}\) with Complete Case Covariate: RQ1 Tackling Heterogeneity  
With accurately measured covariates, we emphasize the necessity of maintaining homogeneity within each risk cluster to determine fair premiums. However, model risk--specifically the heterogeneity issue (RQ1)--emerges when covariates introduce variability, increasing the heterogeneity within these clusters. Assuming that the risk clusters \(j=1,\ldots,J\) are already established, this section presents our foundational hierarchical GLM with varying coefficients, utilizing the partial pooling technique to ensure more homogeneous risk clusters. **Baseline GLM:** For each policy \(h=1,\ldots,H\), we suggest that the claim count \(N_{h}\) follows a negative binomial (NB) distribution with mean \(\xi_{h}\) and dispersion parameter \(\psi\). This setting, along with the assumption that the individual claim amounts on a logarithmic scale \(lnY_{hi},\ i=1,\ldots,N_{h}\), are independent and normally distributed with mean \(\mu_{h}\) and variance \(\sigma^{2}\), is grounded in [PERSON] and [PERSON] (2010) [3]. In short, we represent these two outcome models as follows:  
\[N_{h} \sim\ \mathbf{NB}(\xi_{h},\ \psi)=\frac{\Gamma(N_{h}+\psi)}{N_{h} \Gamma(\psi)}\Bigg{[}\frac{\xi_{h}}{\xi_{h}+\psi}\Bigg{]}^{N_{h}}\Bigg{[}\frac {\psi}{\xi_{h}+\psi}\Bigg{]}^{\psi} \tag{11a}\] \[\widetilde{Y}_{h} \sim\ \mathbf{LogN}(\mu_{h},\ \sigma^{2})=\frac{1}{\widetilde{Y}_{h} \sqrt{2\pi\sigma^{2}}}\exp(-\frac{1}{2\sigma^{2}}\Big{[}ln\widetilde{Y}_{h}- \mu_{h}\Big{]}^{2}) \tag{11b}\]  
With the addition of covariates \(\mathbf{X}=\big{(}\mathbf{X}^{F}:\{\mathbf{x}^{F},\ \mathbf{z}^{F}\},\ \mathbf{X}^{5}:\{\mathbf{x}^{S},\ \mathbf{z}^{5}\}\big{)}\), the covariate effects and the information for risk clusters \(j=1,\ldots,J\) can be integrated into the outcome models via the expectation parameters \(\xi_{h}\) and \(\mu_{h}\) as presented in Equation (11). Specifically, the covariates for claim count (frequency) and claim amount (severity) are represented by \(\mathbf{X}^{F}:\{\mathbf{x}^{F},\mathbf{z}^{F}\}\) and \(\mathbf{X}^{5}:\{\mathbf{x}^{5},\mathbf{z}^{5}\}\), respectively.