## 4 Empirical Study
### Model Validation  
To assess the effectiveness of the proposed models in the following section, the methods outlined below will be utilized:  
Figure 4: Four candidate models—(A) to (D)—for risk premium development. Specifically, Models (B), (C), and (D) need to be thoroughly compared across various error rates \(R_{e_{*}}\)—\(1\%\), \(10\%\), \(40\%\)—in the NDB covariate \(\mathbf{x}^{*}\). * **LPPD:** To assess predictive performance in the Bayesian context, we can employ the Log Pointwise Predictive Density (LPPD) [32], which captures model uncertainty through the complete posterior distribution. LPPD is calculated by averaging the log-predictive likelihood for each data point across posterior samples \(\theta_{1},\ldots\theta_{M}\) as follows: \[\textbf{LPPD}(\bar{Y}_{1},\ldots,\bar{Y}_{H},\textbf{X}_{1},\ldots,\textbf{X}_{ H},\boldsymbol{\theta})=\sum_{h=1}^{H}ln\Big{(}\frac{1}{M}\sum_{m=1}^{M}\textbf{L}( \boldsymbol{\theta}_{m};\bar{Y}_{h},\textbf{X}_{h})\Big{)}\] (39) The likelihood function takes values from 0 to 1 (since it is a probability function), resulting in LPPD values from \(-\infty\) to 0. Multiplying LPPD by \(-2\) yields values similar to Mean Squared Error (MSE), where a perfect fit equals zero and a poor fit results in a large positive value [33]. Note that LPPD emphasizes predictive performance. This is because LPPD not only assesses how well the model fits the training data but also evaluates its generalization to new, unseen data by incorporating parameter uncertainty from the full posterior distribution [32]. * **\(\boldsymbol{D_{KL}}\):** The Kullback-Leibler Divergence \(D_{KL}\) quantifies how one probability distribution differs from another by comparing their entropies \(H[.]\)[34]. Among the competing models, the one with the minimized \(D_{KL}\) is preferred. For our predictive model \(\textbf{L}(\boldsymbol{\theta};\bar{Y}_{h},\textbf{X})\) and target model \(\textbf{P}(\boldsymbol{\theta}^{true};\bar{Y}_{h},\textbf{X})\), \(D_{KL}(\textbf{P},\textbf{L})=H[\textbf{P},\textbf{L}]-H[\textbf{P}]\) for our model can be computed as: \[D_{KL}=-\sum_{h=1}^{H}ln\Big{(}\textbf{L}(\boldsymbol{\theta};\bar{Y}_{h}, \textbf{X}_{h})\Big{)}\cdot\textbf{P}(\boldsymbol{\theta}^{true};\bar{Y}_{h},\textbf{X}_{h})+\sum_{h=1}^{H}ln\Big{(}\textbf{P}(\boldsymbol{\theta}^{true} ;\bar{Y}_{h},X_{h})\Big{)}\cdot\textbf{P}(\boldsymbol{\theta}^{true};\bar{Y}_{ h},\textbf{X}_{h})\] (40) where \(\sum_{h=1}^{H}ln\Big{(}\textbf{L}(\boldsymbol{\theta};\bar{Y}_{h},\textbf{X}_{h })\Big{)}\) is the LPPD discussed previously. We aim to identify which candidate model achieves a larger decrease in \(D_{KL}\) and the extent of that reduction. * **SSPE + SAPE:** Prediction performance can be evaluated by measuring the discrepancy between predicted and observed values using the Sum of Square Error (SSE) criterion. Specifically, the Sum of Square Prediction Error (SSPE) and the Sum of Absolute Prediction Error (SAPE) can be used to assess different facets of accuracy. SSPE focuses on squared differences between predicted values \(g(\textbf{X}_{h})\) and actual values \(\bar{Y}_{h}\), while SAPE calculates the absolute differences across all observations \(h=1,\ldots H\): \[\text{SSPE:}\ \sum_{h=1}^{H}\big{(}g(\textbf{X}_{h})-\bar{Y}_{h} \big{)}^{2}\] (41a) SAPE: \[\sum_{h=1}^{H}\big{|}g(\textbf{X}_{h})-\bar{Y}_{h}\big{|}\] (41b) SSPE and SAPE assess prediction performance differently. SSPE penalizes large deviations more heavily, while SAPE treats all deviations equally by focusing on absolute differences. Given the skewed nature of our outcome \(\bar{Y}_{h}\), where outliers may be common, SAPE is preferred. This approach considers each data point equally important, making it unnecessary to disproportionately penalize larger errors, particularly when we are concerned with potential outliers [13].