## 4 Discussion  
This realistic case study has shown that a Gaussian discrepancy term, with null mean and unknown variance, is an effective choice to tackle the model inaccuracy rising for low levels of degradation when the aim is inference of the system's initial stiffness. At the same time, it has been proven that this kind of discrepancy still works fine for low levels of damage even if the input is rescaled up to 1 g. Consequently, the discrepancy term adopted can be judged suitable to cover model errors with increasing values of the PGA. However, higher levels of PGA introduce higher uncertainties in the model prediction. It has been shown that increasing the amount of degradation jointly to the level of damage leads to inaccurate predictions for those models that are inconsistent with the original physical BWBN model and more uncertainty also arises in the latter. This can be appreciated, for example, in the velocity response prediction of Model 3 (Figure 15). This is the model that leads to the worst model prediction for high level of degradation, owing to the presence of a greater number of overestimated outliers. Figure 14: (**a**) Kernel density estimations (KDE) of the initial stiffness model parameter \(k_{i}\) and; (**b**) of the discrepancy variance \(\sigma^{2}\) for each model (medium level of degradation: PGA \(\approx\) 9 m/s\({}^{2}\), stiffness reduction up to 80% of the initial one). (**c**) Boxplot representations (minimum, first quartile, median, third quartile and maximum) of the posterior samples for the model parameter \(k_{i}\) and; (**d**) \(\sigma^{2}\). The horizontal and vertical dotted lines in the plots represent the calibrated reference value of the initial stiffness. Figure 15: Velocity response prediction of the BWBN Model 3 for high level of degradation (PGA of 9 m/s\({}^{2}\), stiffness reduction up to 80% of the initial one). What has been determined can be easily read from the graphs in Figure 16. The latter depicts the MSE error over the time of the velocity response prediction for each case. Specifically, Figure 16a-c display the variation of MSE computed for all the post-predictive model runs, for each model and for each level of degradation (low, medium and high). Instead, the comparison between the average MSE computed on all the post-predictive models runs versus the MSE of the MAP estimate prediction at different levels of degradation, is presented in Figure 16d. From the inspection of latter, we can conclude how the error in prediction increases as the level of degradation increases. This leads to more uncertain predictions. At last, for those cases inconsistent with the physics model that presents from a medium to high level of damage, it can be concluded that the simple discrepancy function adopted is not sufficient to embody all the model errors. This means that the choice of the discrepancy function should be linked to the level of damage and, in actual practice applications, to the level of excitation. Thus, to summarize the considerations on the use of Bayesian updating and the uncertainty quantification framework, one can state that the model errors and the observation errors can be taken into account by adding a discrepancy term in the model calibration process. This brings benefits in terms of calibration outcomes (i.e., in addition to good time-series fitting, the order of the parameters is close to the real value). However, for some cases, the Bayesian approach within the uncertainty quantification framework could bring trivial results in terms of model parameter values. This especially occurs if the discrepancy model distribution is not chosen properly. The findings of this paper show that a good way to choose the discrepancy model distribution could be to study potential  
Figure 16: Boxplot representations (minimum, first quartile, median, third quartile and maximum) of the mean square error (MSE) computed on all the post-predictive models runs.