## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
In practice, changes in the moment of inertia may occur when the geometric configuration of certain regions deviates from the original design, or due to loosening of diagonal bracing in some subregions. Such changes can indeed lead to significant variations in the moment of inertia, potentially by as much as 30% or more, as shown in references [37; 76; 77]. It is important to emphasize that when analyzing a real structure, it would indeed be necessary to carefully consider the actual types of damage in order to ensure that the damage identification has true physical meaning. It is worth mentioning that the support spring stiffness is also considered an updating parameter. In Table 3, we account for a 20% increase in the supporting stiffness, which is not necessarily a result of structural damage. It is indeed possible for the spring stiffness to increase under certain conditions. For instance, this could occur due to repairs or reinforcements in the support structure, changes in the boundary conditions, the addition of stiffer materials, or structural modifications that lead to a more rigid connection. The design of the stiffness variation in the supporting spring in Table 3 is based on studies [60; 61; 78; 79]. #### 4.3.2 Results

To implement DRAM, TMCMC, and DREAM, the algorithm settings are identical to those given in Sections 4.1 and 4.2, except for a setting where a sample size of 15,000 with fewer parameters to be updated was designed. The lower and upper limits of the 15 parameters are set as 0.5 and 1.5, respectively. The trace plots of 15 stiffness parameters for three algorithms are shown in Figures 18-20. It can be seen that the Markov chains of all parameters obtained by DRAM appear to be noticeable segments with stagnation and the phenomenon of'standing still', indicating a very low acceptance rate during sampling and leading to the mis-estimation of stiffness parameters. Turning our attention to Figures 19 and 20, the samples of all parameters fluctuate in the vicinity of the true values (red dashed lines), demonstrating a satisfactory convergence using TMCMC and DREAM. It was also found that TMCMC reaches convergence faster than DREAM, mainly because the intermediate distribution in TMCMC reaches the target posterior at an early stage. Figure 21 displays the trace plots of five Markov chains in DREAM in terms of parameters at damaged substructures. All chains have consistent convergence to the ground truth, proving robust and reliable parameter estimation using DREAM. Table 1.3 Trace plots by DRAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations. Figure 22 shows the marginal distributions of parameters at damaged substructures obtained by TMCMC and DREAM. As observed, diagonal posterior distributions of the two algorithms are well matched with each other and cover the true values, implying that both algorithms successfully achieve damage localization and quantification but also provide valuable confidence in damage detection. The estimation errors in parameter identification by DRAM, TMCMC, and DREAM are reported in Table 1.4 It was found that DRAM gives completely incorrect parameter estimation due to sampling stagnation and sensitivity to initial values, resulting in incorrect damage detection. Conversely, both TMCMC and DREAM exhibit desirable damage detection, probably because of the lower number of to-be-updated parameters compared to previous examples where DREAM always performs best. DREAM slightly outperforms TMCMC; e.g., it has a maximum error of 2% and 6%, respectively. Regarding the computational costs, DRAM only takes around 4 min for sampling but fails to correctly identify parameters. TMCMC takes around 25 min, and DREAM takes the longest, at around 30 min, as multiple Markov chains are concurrently running on the samples. Table 1.5 Trace plots by TMCMC: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value).