## 4 Results and Discussions  
Figure 9 shows the application results of the PRE method in Scenario 1 (no stiffness reduction). It consists of the figures indicating the probability density distributions of the estimated mechanical parameters of the VBI system. Each vertical axis indicates the probability density of each estimated parameter, while each horizontal axis denotes the normalized value. The red and blue bars in each histogram represent the prior and posterior distributions, respectively. The posterior distributions are obtained by updating the particles \(\boldsymbol{X}_{i}^{(k)}\) with the MCMC-powered PRE method. The left and right show the vehicle's and the bridge's mechanical parameters. Noted that the coordinate \(\boldsymbol{X}_{i}^{(k)}\) does not include the vehicular body mass: \(m_{s1}\) and \(m_{s2}\) but \(d_{1}\). Since \(M\) and \(D\) are fixed, \(m_{s1}\) and \(m_{s2}\) are automatically decided when \(d_{1}\) is given. Because of the distance from G: \(d_{1}\) is randomly assumed to range from 0.8 to 1.2 times the correct value, the normalized prior probabilities of \(m_{si}\) in the initial step do not distribute from 0.8 to 1.2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Process** & **Parameter** & **Value** \\ \hline Dynamic simulation & The Newmark-\(\beta\) method & \(\gamma=1/2\) \\ and numerical integration & Time increment & \(\Delta t=0.001\) \\ \hline Monte Carlo Markov chain & Particle number & 500 \\ method & Step number & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The parameters of the PRE method with MCMC simulation. According to Figure 9, the updated distributions peak around 1.0, which means that the PRE method can estimate these mechanical parameters. The posterior probability density distributions of all parameters except bridge damping coefficients \(\alpha\) and \(\beta\) are improved from the prior information by the MCMC process. Figure 10 also shows the correct and estimated road unevenness \(R(x)\), \(\hat{R}_{1}(x)\) and \(\hat{R}_{2}(x)\) by the PRE method using the optimal particle \(\textbf{X}_{i}^{(600)}\) of which \(J\) value is minimum. In this study, it is confirmed that when substituting correct values instead of assumed values, \(J\) becomes zero. If the updated values are close to the correct values in the PRE process, the difference between the estimated road unevenness functions must be almost zero. Figure 9: The application results of the PRE method in Scenario 1: these figures indicate the prior and posterior probability density distributions of the estimated mechanical parameters. The blue bars represent the prior information assuming uniform probability distributions. The red bars represent the posterior information updated by the MCMC process. The left and right indicate the vehicle and bridge mechanical parameters, respectively. However, there is no guarantee that the matching two estimated unevenness functions are also close to the correct waveform. However, according to [PERSON] et al. [27], the Monte Carlo-based process reproduces the correct road unevenness, while Figure 10 also shows the high estimation accuracy of road unevenness \(R(x)\). This means that the same tendency in the previous study [27] is confirmed; when the estimated road unevenness functions match, both also match the correct one. Their tiny errors and differences are shown in Figure A4, Appendix A. Figures 11 and 12 are the results obtained from the case of Scenario 6. In this scenario, \(EI_{4}\) is reduced in numerical simulation. According to these figures, the PRE method can estimate the stiffness reduction of the stiffness decrease element. This result also suggests that the PRE method can update the parameters to the correct values if wrong prior information is given. However, the estimated stiffness \(EI_{4}\) distributes higher than the correct value of 0.6 in this scenario. This overestimation should be improved in future works because it can lead to a severe accident.