## 3 Estimation of Model Parameters
### Adaptive MCMC  
As outlined above, MCMC is a quintessential parameter estimation method within a Bayesian framework. The fundamental concept involves constructing a Markov chain using sampling techniques to ensure that the steady-state distribution aligns with the posterior distribution of the target parameter. Adaptive MCMC enhances this process by dynamically fine-tuning the sampling parameters, thereby boosting sampling efficiency [44].  
First, determine the posterior distribution function of the parameters to be estimated. According to the Bayesian criterion, given the observed data \(\mathbf{Data}_{k}=\big{\{}(t_{k,m},x_{k,m})\big{\}}\), the posterior distribution \(p(\mathbf{\gamma}_{k}|\mathbf{Data}_{k})\) of the parameters to be estimated \(\mathbf{\gamma}_{k}=\{\beta_{k},\theta_{k},\varphi_{k}\}\) is proportional to the product of the prior distribution \(p(\mathbf{\gamma}_{k})\) and the likelihood function \(p(\mathbf{Data}_{k}|\mathbf{\gamma}_{k})\), as expressed in Equation (11):  
\[p(\mathbf{\gamma}_{k}|\mathbf{Data}_{k})=\frac{p(\mathbf{Data}_{k}|\mathbf{\gamma}_{k})\cdot p (\mathbf{\gamma}_{k})}{p(\mathbf{Data}_{k})}\propto p(\mathbf{Data}_{k}|\mathbf{\gamma}_{k}) \cdot p(\mathbf{\gamma}_{k}), \tag{11}\]  
where the prior distribution \(p(\mathbf{\gamma}_{k})\) in this study is a multivariate truncated normal distribution with a mean of \(\mathbf{\mu}=\big{(}\mu_{\beta},\mu_{\sigma},\mu_{\theta}\big{)}\), a covariance matrix of \(\mathbf{C}_{3\times 3}\), and truncation interval \(\mathbf{LU}=\{\beta_{k}>0,\sigma_{k}>0,\theta_{k}\geq 1\}\). This study used the MLE-based results of the parameters for the deterioration models of all segments. The mean and covariance of these results will respectively serve as the \(\mathbf{\mu}\) and \(\mathbf{C}_{3\times 3}\) for the prior distribution. According to Equation (8), the likelihood function is as follows:  
\[p(\mathbf{Data}_{k}|\mathbf{\gamma}_{k})=\prod\
olimits_{m=2}^{M_{k}}\frac{1}{\sqrt{2 \pi\sigma_{k}{}^{2}\Delta\Lambda(t;\theta_{k})_{m}}}\text{exp}[-\frac{\big{(} \Delta x_{k,m}-\beta_{k}\Delta\Lambda\big{(}t;\theta_{k})_{m}\big{)}^{2}}{2 \sigma_{k}{}^{2}\Delta\Lambda(t;\theta_{k})_{m}}], \tag{12}\]  
Secondly, the sampling method and adaptive adjustment rules were determined. Considering the complexity of sampling from the posterior distribution and the correlation between parameters, this study adopts the more flexible Metropolis-Hastings algorithm for the sampling process. For the adaptive adjustment rules, based on the discussion in the literature [44], the study employs the optimization criterion of aiming to achieve an optimal acceptance rate range of \(0.2<\alpha^{*}<0.3\) to continuously adjust the covariance matrix of the proposal distribution during the burn-in period. The adjustment coefficient \(\lambda\) is determined by conducting parameter estimation on randomly selected track segments and comparing the convergence under different coefficient values. The length of the burn-in period was set to 50% of the total number of samples. Samples generated during the burn-in period were not used for subsequent statistical analyses [44]. In addition, considering that the input of the initial samples has a significant impact on the convergence speed of the sampling process and the accuracy of the parameter estimation results, the sampling algorithm was executed using the MLE-based results as the initial values for the parameters to be estimated. Algorithm 2 presents the pseudocode for the adaptive MCMC algorithm used to estimate the model parameter \(\mathbf{\gamma}_{k}\).