## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
Table 1.6 Trace plots by DREAM: (**a**) parameters at undamaged locations; (**b**) parameters at damaged locations (red dashed line is true value). Table 1.7 Trace plots of five Markov chains by DREAM: (a) \(\theta_{1}\); (b) \(\theta_{3}\); (c) \(\theta_{6}\); (d) \(\theta_{8}\); (e) \(\theta_{10}\); (f) \(\theta_{11}\); (g) \(\theta_{12}\); (h) \(\theta_{15}\). Table 4 presents a comparative analysis of DRAM, TMCMC, and DREAM and evaluates their performances in terms of accuracy, computational cost, and complexity across a shear building, continuous beam, and pedestrian bridge. In terms of accuracy, DRAM exhibits the lowest values across all cases, indicating that while it may be efficient, it struggles with accurate damage detection and parameter estimation. TMCMC provides medium accuracy, suggesting it offers a balanced approach. While there are noticeable errors between the estimated and true parameters, the level of damage detection remains acceptable. DREAM achieves the highest accuracy, excelling in both damage detection and parameter estimation, offering the precise identification of structural damage. Regarding computational cost, DRAM is the most computationally efficient, requiring the least amount of time. However, this efficiency comes at the cost of lower accuracy. TMCMC falls in the middle regarding computational cost. It strikes a balance between accuracy and computational efficiency. DREAM is the most computationally costly. This higher computational cost correlates with its superior accuracy. Table 1.8 Estimation error by DRAM, TMCMC, and DREAM. Table 1.9 Marginal distributions of model parameters. The complexity of the algorithms also plays a significant role in their performance. Both DRAM and TMCMC operate using a single chain for sampling, which contributes to their lower computational cost but limits their accuracy. DREAM, on the other hand, utilizes multiple chains, enhancing accuracy but also increasing computational cost. The use of multiple chains allows DREAM to explore the parameter space more thoroughly. Each chain can sample from different regions of the parameter space, reducing the risk of becoming trapped in local optima and increasing the chance of finding the global optimum. This thorough exploration leads to higher accuracy in damage detection and parameter estimation. However, running multiple chains simultaneously requires significantly more computational resources, resulting in higher computational costs. ### Discussion

Three MCMC algorithms are applied to three engineering structures with increasing complexity. This Section discusses the performance of each algorithm in the context of damage detection based on a comparative study. DRAM has a very easy-to-understand algorithm mechanism and a relatively simpler implementation compared to TMCMC and DREAM, leading to the least computational cost. The settings for implementation include the design of the DR stage, e.g., scaling factor and number of the DR stage, and the design of the AM stage, e.g., a non-adaptation period, as well as initial samples. Through all application cases, it can be seen that DRAM is ineffective in sampling (e.g., sampling stagnation) for high-dimensional problems, mainly owing to the random walk principle. The DRAM would encounter arduous convergence challenges for the target posteriors with complex geometry. DRAM's capability is greatly limited by the choice of DR and AM or other tuning parameters. A more appropriate DR and AM strategy design would be more beneficial to parameter estimation. Unfortunately, to the best of the authors' knowledge, there are no comprehensive guidelines for DR and AM. A possible remedy is to tune the DR and AM processes in a trial-and-error manner with user interaction in practice. In addition, it was observed that the selection of initial samples significantly affects the DRAM's performance on parameter estimation, as shown in Figures 3 and 11.