## 5 Discussion
### Future Work  
There are several concerns regarding our hierarchical GLM risk premium modeling equipped with Gustafson correction. 1. **Dimensionality:** In our current analysis, we have limited our focus to two covariates--one binary and one continuous--for the sake of simplicity. However, it is imperative to explore more intricate datasets that encompass a greater variety of covariates. In the Bayesian framework, as the number of covariates increases, the complexity of the likelihood components--essentially the models representing these covariates--also expands. This escalation may introduce additional noise or unobserved structures that could compromise the stability of the resulting predictive distributions [22]. Therefore, conducting further investigations into the challenges posed by high-dimensional covariates within the hierarchical GLM framework presents an opportunity for enhancing model robustness and accuracy. 2. **Reliability of risk clusters:** As a parametric Bayesian approach, fixed risk clusters in the hierarchical GLM framework have several downsides that merit further exploration. Above all, if the predetermined risk clusters are not entirely accurate, this can lead to additional bias and possibly reduce model validity [21]. However, when the fixed clusters are indeed correct, the partial pooling mechanism within the hierarchical GLM framework demonstrates its superiority, effectively leveraging the structure of the data while improving parameter estimates. Thus, future research should investigate more flexible Bayesian clustering methods that allow for dynamic cluster adjustments. 3. **Granularity of error rates:** In the development of simulation data, we sought to determine the error rate at which the Gustafson correction becomes ineffective. Our findings indicate that for error rates \(R_{e_{x}}\) below 0.01, the application of the correction is generally unwarranted, as model risk remains minimal under such conditions. In the range of error rates from 0.10 to 0.50, the correction demonstrated its utility. Specifically, it was effective when the scaling factor \(0.5\leq\zeta\leq 0.7\), as the hidden relation between \(\widehat{\lambda}^{2}:V(\mathbf{x}^{*}|\mathbf{z})\) and \(\mathbf{\mathfrak{T}}^{2}:V(\mathbf{x}^{*}|\mathbf{x})\) in the LGPIF dataset is explained by the range of \(\zeta\). However, we hypothesize that at error rates exceeding 0.50, the correction may encounter considerable limitations in its effectiveness, despite our identification of the optimal range for \(\zeta\). It is noteworthy that this specific scenario has not yet been explored in our study, indicating a potential avenue for future research utilizing different datasets. * **Scalability of posterior simulation:** Lastly, we proposed an examination of the scalability of posterior simulations utilizing our Gibbs sampler. Our empirical analysis, particularly with the LGPIF dataset, demonstrates that the hierarchical GLM framework maintains stable performance with sample sizes up to \(H\leq 2000\). However, the impact of larger sample sizes, specifically those exceeding 10,000, has not been addressed within the scope of this paper. As the volume of data increases, it raises critical questions regarding the computational efficiency of our hierarchical GLM framework, particularly given the heightened demand for computational resources and the potential for performance degradation as noted in the studies of [PERSON] et al. (2020) [38]. This aspect becomes increasingly significant in contexts where insurance loss data are anticipated to accumulate over time, necessitating a thorough exploration of the framework's capacity to scale effectively while maintaining performance. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15010210/s1](https://www.mdpi.com/article/10.3390/app15010210/s1)]([https://www.mdpi.com/article/10.3390/app15010210/s1](https://www.mdpi.com/article/10.3390/app15010210/s1)). All authors contributed substantially to this work--Conceptualization, M.K.