## 4 Results and Discussions
The restoring force and damping force generated by each suspension are internal. Thus, these two acting internal forces can be eliminated from Equation (1). The accuracy improvement of these values will be a technical issue of future work. On the other hand, the standard deviations of the bridge stiffness at both edges \(EI_{1}\) and \(EI_{2}\) also tend to be higher. Since the bridge vibrations at both ends should be smaller than others, it becomes difficult to evaluate their influence on vehicle responses. While the reduced stiffness values tend to be overestimated, the stiffness next to the damaged element tends to be underestimated. Using this characteristic of the PRE method, there is a feasibility of detecting edge stiffness reduction. The Rayleigh damping coefficients \(\alpha\) and \(\beta\) are estimated with low accuracy. This means that the effectiveness of \(\alpha\) and \(\beta\) to the dynamic responses of the vehicle is minimal. In Scenario 4, Scenario 5, and Scenario 6, the flexural rigidity of the stiffness decrease element \(EI_{4}\) tends to be lower than that of Scenario 1, while \(EI_{3}\) and \(EI_{5}\) also decrease despite their soundness. Similarly, in the cases of Scenario 7, Scenario 8, and Scenario 9, the bending stiffness values of the stiffness decrease elements \(EI_{5}\) and \(EI_{6}\) can be estimated, while \(EI_{4}\) is also underestimated. In the cases of Scenario 10, Scenario 11, and Scenario 12, it is difficult to recognize the stiffness reduction tendency from the posterior distribution of \(EI_{7}\), while the underestimated value of \(EI_{6}\) can be used for the detection of stiffness reduction in Scenario 12. Table 8 shows the estimation accuracy of the PRE method. The accuracy is evaluated by Equation (23). Values closer to 1 indicate better accuracy. The bold style indicates the estimated values of stiffness decrease elements. The estimation accuracy of the stiffness decrease elements and near elements tend to be lower. This low accuracy can be used to detect bridge stiffness reduction but not to estimate it. The problem of stiffness overestimation and underestimation may be due to the limitation of the update efficiency of the MCMC scheme. Changing the optimization scheme may improve accuracy. It should be noted that the accuracy of \(EI_{1}\) of Scenario 1 is 0.98. It seems to be accurate, but the standard deviation of \(EI_{1}\) shown in Table 6 is 0.23, which is bigger than others (ranging from 0.1 to 0.2). Therefore, it can be said that the estimated \(EI_{1}\) in this scenario is less reliable than the others. The performance of the inspection scheme should be evaluated by accuracy and reliability. According to Table 7, the accuracies of \(EI_{3}\) and \(EI_{4}\) are 0.79 and 0.86, respectively. While their correct values are 1.0 and 0.4, the average of \(EI_{3}\) and \(EI_{4}\) shown in Table 6 are 0.79 and 0.54. This means that the estimation accuracies of \(EI_{3}\) and \(EI_{4}\) are similar, but \(EI_{3}\) is underestimated, while \(EI_{4}\) is overestimated. These results of applying the PRE process are characterized by the ability to estimate all three VBI system elements: vehicles, bridges, and road surfaces. The comparison between the performance of the PRE method and existing schemes should be considered. [PERSON]'s scheme [23], based on a one-tractor-two-trailers system, can estimate the bridge mode shape with perfect accuracy. The estimation accuracy of bridge stiffness provided by the PRE method is not as high as [PERSON]'s scheme [23]. However, [PERSON]'s method cannot extract road roughness information but erase it. The vehicle's dynamic characteristics are analyzed but not identified. The PRE method assumes to be applied to inspect vehicle conditions in the future, while [PERSON]'s method is not concerned with the state of the two monitored vehicles. However, [PERSON]'s method is superior because it can be expected to have the same accuracy as a numerical simulation, even when used in an actual environment.