## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
Previous studies have demonstrated the feasibility of the parallel computing of MCMC [81]. ### Practical Aspect

In terms of selecting the three advanced MCMC methods for practical application, we suggest employing a sequential implementation strategy for posterior investigation and cross-validation. Begin with the DRAM algorithm. DRAM is advantageous for its computational efficiency and straightforward implementation, making it ideal for preliminary parameter estimation and damage detection. It serves as an excellent starting point, allowing for quick and efficient initial assessments. Following the preliminary assessment with DRAM, we advise transitioning to TMCMC or DREAM for further refinement and validation. TMCMC is recommended for its balanced approach, offering a moderate level of accuracy with reasonable computational costs, thereby serving as an effective means to verify and build upon the initial DRAM results. On the other hand, DREAM should be employed when the highest level of accuracy is required. Although DREAM is computationally intensive, its use of multiple chains enables the thorough exploration of the parameter space, ensuring precise damage detection and parameter estimation, which is particularly beneficial for complex structural analyses. In summary, initiate the process with DRAM for rapid and efficient preliminary analysis. Then, proceed with TMCMC or DREAM to achieve more detailed and accurate validation. This sequential approach ensures a robust and comprehensive examination of posterior distributions, ultimately providing reliable and precise damage detection results. It is worth mentioning that the scalability of the discussed MCMC techniques was investigated in three different structural types with increasing complexity, ranging from a 40-DOF shear building to a 274-DOF pedestrian bridge. The computational times for each method are all less than an hour. However, for real-world large-scale engineering structures, FE models typically involve hundreds of thousands of elements and nodes. Evaluating such forward models or likelihood functions can become computationally intensive, with each model potentially taking several minutes to run. Given that MCMC methods typically require around 104 evaluations to ensure convergence, the direct application of Bayesian model updating for these large-scale structures becomes computationally prohibitive. To address these challenges, developing surrogate models of the expensive forward model presents a viable solution. Surrogate models, such as Kriging models [82], polynomial chaos expansions [83], and neural networks [84], can approximate the behavior of the forward model with significantly reduced computational cost. By employing these surrogate models, the evaluation process is accelerated, making the application of Bayesian model updating in large-scale structures both feasible and practical. While the scalability of MCMC techniques is limited by computational resources and time in the context of large-scale systems, the integration of surrogate models offers a promising pathway to overcome these limitations and facilitate efficient Bayesian model updating for complex engineering structures. ## 5 Conclusions

Bayesian model updating is well-known for parameter estimation and structural damage detection in engineering structures, as confidence in damage detection can be directly and reasonably provided given the posterior distributions. In Bayesian model updating, MCMC algorithms are used as computational tools to sample from the target posteriors. The performance of MCMC algorithms heavily relies on their own mechanisms. Various MCMC algorithms have been developed and prospered to enhance the accuracy and reliability of the sampling process. In this comparative study paper, three advanced MCMC algorithms, namely, DRAM, TMCMC, and DREAM, were reviewed and compared.