## 1 Introduction
With this scheme, all information about the VBI (vehicle-bridge interaction) system can be reproduced only from vehicle vibration and position data. However, the applicability of this method to bridge inspection has not been thoroughly investigated. In the general bridge inspections, changes in appearance and hammering sounds are collected mainly to estimate stiffness changes. Thus, the parametric changes in bridge stiffness should be methodologically considered in this paper. The vehicle's vertical acceleration vibrations and position information are measured in the PRE method. These measured data and assumed mechanical parameters are used to estimate two road profiles of the front and rear wheels in the process. Both estimated road surface unevenness functions should match when the vehicle moves straight ahead. However, the estimated road unevenness in the initial step rarely matches because the mechanical parameters are randomly assumed. Therefore, the sum of the squared error between the road unevenness estimated at the front and rear wheels is adopted as the objective function to find the optimal mechanical parameters. The PRE method searches for the mechanical parameters minimizing this function. It is noted that the wheelbase and gross weight of the vehicle are assumed to be known because the obtained solution could be indeterminate. Because these two known parameters are easy to be measured, the PRE efficiency does not decrease. According to the previous studies of the PRE method [30,31], the converged parameters can match the correct values. The same tendency can be confirmed in road profile estimation using the random parameter assumption [27]. There are several methods of searching for the optimal solution of the mechanical parameters. [PERSON] et al. [26] adopted GA to identify the parameters, while [PERSON] et al. [27] applied cross-entropy optimization. The success of these existing similar studies indicates that a scheme based on Monte Carlo simulation can be applied to the PRE method. At the same time, it should be noted that the difference between this paper and the existing studies [26, 27, 28, 29] lies in bridge system identification. The PRE method needs to search for the optimal solution in an ample unknown multidimensional space. Thus, as the first step, this paper adopts the MCMC (Monte Carlo Markov chain) model's so-called random walk for the optimization process. The damage detection performance of the PRE method should be evaluated quantitatively to verify its applicability. In previous studies [30; 31], structural changes have been mainly considered as stiffness reduction; according to the findings of these studies, the PRE method can estimate the location and magnitude of stiffness reduction. However, whether the stiffness reduction can be detected in places with slight vibration, such as at a bridge end, is unclear. Furthermore, the estimation sensitivity to a slight decrease in stiffness should be investigated. Another current study [32] provides the application results of the PRE method to the data obtained from the field experiment and concludes that there are still many difficulties in implementing this scheme using actual data. Therefore, this study aims to verify the proof-of-concept of the MCMC-powered PRE method for estimating bridge stiffness reduction. The dynamic responses of the VBI system are numerically simulated while changing the bending stiffness of the bridge model. Twelve patterns with different locations, sizes, and magnitudes of stiffness reduction are prepared. Since the MCMC process gives the estimation results as probability density distributions, the means and standard deviations of the estimated parameters are used as indices to quantitatively express the accuracy and reliability of the PRE method, respectively.