## 5 Discussion
### Influence of Model Selection Criteria on Predictions and Model Complexity  
We compared the predictive performance of models tuned using three model selection criteria: CV, AIC, and BIC. Our findings showed that the performance of these criteria varies according to sample size, SNR, and the correlation among covariates. In scenarios with limited information, BIC generally led to worse prediction accuracy than AIC and CV. This is because BIC's strong penalty for model complexity often results in models with too few variables (Figure 12), leading to higher prediction errors. These findings corroborate those from other related studies on variable selection [64]. In contrast, when sufficient information was available, BIC often outperformed AIC and CV in accuracy. However, this was not the case for the lasso model, where BIC consistently resulted in inferior predictions. This likely stems from lasso's use of a single tuning parameter to simultaneously control both variable selection and shrinkage [16]. Since BIC tends to favor larger tuning parameters, it can cause excessive shrinkage, leading to suboptimal model performance. Initial estimates in ALASSO and NNG, as well as the auxiliary parameter in RLASSO, help reduce the amount of shrinkage applied to coefficients. This can improve both model selection and prediction accuracy [16; 40]. Furthermore, in scenarios with several small effects (beta-type B), BIC-tuned models performed poorly, even when the data contained sufficient information (Figure 13), as BIC tends to eliminate small effects and thereby underfit the model. This observation aligns with findings by [PERSON] and [PERSON] [35], who reported that BIC performs best when only large effects are present, whereas AIC is more effective when both large and small effects exist. AIC and CV performed similarly in most scenarios, which is consistent with their shared objective of selecting models that generalize well to unseen data [35]. [PERSON] [65] provided a theoretical basis, showing that under certain conditions, the model selected by leave-one-out CV (a special case of k-fold CV) asymptotically converges to the model selected by AIC. Regarding model complexity, BIC generally selected models with fewer variables than AIC and CV, as expected due to its more stringent penalty for model complexity. AIC and CV selected a comparable number of variables. It is important to note that there is a link between model selection criteria and model selection tests (i.e., hypothesis tests comparing different models). [PERSON] and [PERSON] [66] demonstrated that using model selection criteria to select models is akin to selecting a model at a certain significance level. Under certain assumptions, they derived approximate significance levels for different model selection criteria. For AIC, the asymptotic significance level for including a variable in a model corresponds to 0.157, derived from the upper tail area of the chi-squared distribution with one degree of freedom, where the cutoff value is \(a=2\). In contrast, for BIC, the significance level is a function of the sample size, where \(\text{a}=\ln(n)\); for example, the significance levels for \(n=100\) and \(n=400\) are approximately 0.032 and 0.014, respectively. These smaller significance levels explain BIC's tendency to select simpler models compared to AIC. Impact of a High Proportion of Noise Variables on Prediction Accuracy and Model Complexity of Approaches  
We assessed the robustness of variable selection methods in the presence of a relatively large number of noise variables. The results were similar to those obtained with a lower proportion of noise variables. In contrast, previous simulation studies [9; 21] have reported that subset selection methods are sensitive to the proportion of noise variables, which does not align with our findings. This discrepancy is likely because we added only 15 noise variables, which may have been too few to show substantial differences in performance across methods.