## 6 Calibration of the [PERSON] \(\mathbf{I_{V}}\) Method for Campi Alto di Norcia  
Stating the results previously presented, the Vicente method is the one showing more performing results and, besides, including aggregate considerations; with the aim of achieving an \(\mathbf{I_{V}}\) method suitable for damage prediction in Campi Alto di Norcia, a calibration procedure was performed to opportunely modify the weights associated to the different parameters achieving a good agreement between observed and theoretical damage scenario. The issue consisted of relating the distribution of real/observed damages to the vulnerability assessments carried out using parameters with fixed weights and classes, translating--in this specific application--to a problem with 67 linearly independent equations (representing the damage of the buildings surveyed) and 14 unknown variables (the weights used for each building) that cannot be solved at a deterministic level but require a statistical approach where the evidence of a true state (i.e., damage detected) is expressed in terms of degrees of probability. A Bayesian linear regression was therefore adopted. The calibration of an \(\mathbf{I_{V}}\) method was tackled as an inference problem; the probability of having a particular event E (the damage) given the occurrence of A (the vulnerability) was expressed by Equation (9), representing the basic idea used for this calibration. \[\mathrm{P}(\mathrm{E_{i}}|\mathrm{A})=\frac{\mathrm{P}(\mathrm{A}|\mathrm{E_{ i}})\cdot\mathrm{P}(\mathrm{E_{i}})}{\mathrm{P}(\mathrm{A})} \tag{9}\]  
The statistical model combined some a priori knowledge of the weights with the observed data points through Equation (10). The weights (\(\theta\)) were considered as a random vector, following the so-called prior distribution \(\pi\). After checking the reliability of the first distribution, a posterior distribution \(\pi(\theta|\ y)\) was evaluated through a likelihood function \(\pi\) (\(y|\ \theta\)), representing the probability of observing the data (\(y\)) given a certain set of weights theta (\(\theta\)) [41]. \[\pi(\theta|\mathrm{x})=\frac{\pi(\mathrm{x}|\theta)\pi(\theta)}{\pi(\mathrm{x })} \tag{10}\]  
The posterior distribution represented then an update of the prior one correcting the initial estimation. The Markov Chain Monte Carlo simulation sampling method (MCMC) was used for computing the solution. According to an iterative process, a proposed new distribution was accepted or rejected until the solution converged within a given probability [42]. Figure 17 summarizes the procedure followed to define the weight distribution describing the real damage scenario detected in the case study. Figure 17: Procedure followed for the identification of the weight distribution. The analysis of the correlation between the weight distributions and the real damage detected, and its subsequent improvement, was made using MATLAB-based software framework called UQLab [43]. Figure 18 and Table 7 compare the calibrated and non-calibrated weights used in the Vicente method for vulnerability assessment and damage scenario evaluation. The reliability of the I\({}_{V}\) method was improved, reducing the mean relative error between real and theoretical damage from 0.85 to 0.49, considering the direct measure (i.e., each SU individually with no uncertainty). Considering the full sample range of the calibrated vulnerability assessment, the mean relative error between real and theoretical damage shifted from 0.58 to 0.53. Since the calibrated IV values were less concentrated around an average value and more distributed along the damage curve, the SD did  
The weights' calibration was performed through the damage scenario of a case study hit by a seismic event with a specific intensity (I\({}_{\text{EM}998}\) = 7.65 \(\approx\) 8.0): this means that the vulnerability assessment of a historical centre was carried out with the calibrated weights only with a seismic intensity similar to that used.