## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
Their concepts and basic mechanisms were introduced, and a comparative study of the damage detection of three different engineering structures with increasing complexity was also thoroughly investigated. DRAM takes advantage of DR and AM strategies, but its effectiveness is still restricted to high-dimensional problems and its strong dependence on the initial values. Of all the examples in this study, DRAM performs most poorly in regard to damage detection. The concept in TMCMC is that a series of intermediate distributions is used to sample from the posterior rather than directly sampling. The results show TMCMC that enables the achievement of desirable parameter estimation for moderate-dimensional problems, e.g., 15 dimensions, while, from the application examples presented, it can be seen that DREAM has the strongest capability in estimating parameters, given that it consistently sampled the target posteriors in the 15, 30, and 40 dimensions at a fairly accurate level. This is attributed to the interactions between Markov chains and randomized subspace sampling. However, it comes with a trade-off between sampling efficiency and computational cost. The results highlight that DRAM, while computationally efficient, struggles with high-dimensional posteriors due to sampling stagnation and sensitivity to initial samples, making it less suitable for complex structural models. TMCMC performs better in moderately high dimensions but degrades in accuracy for very high-dimensional problems, which can lead to misleading damage detection results. DREAM, on the other hand, consistently provides the most accurate and reliable results, particularly for high-dimensional problems, thanks to its use of multiple Markov chains and differential evolution. In addition, DREAM's higher computational cost, due to the large sample size required, can be mitigated through parallel computing. In conclusion, the study's results suggest that while DRAM and TMCMC offer certain advantages in computational efficiency and simplicity, DREAM stands out for its ability to handle complex, high-dimensional posteriors more reliably. This makes it a strong candidate for use in Bayesian model updating and structural damage detection where precision and robustness are critical. Future work could explore optimizing computational efficiency in DREAM through parallelization or other advanced techniques, making it more applicable to real-world scenarios where high dimensionality is common. It should be noted that, in this study, all the application examples are numerical, with the final example involving a benchmark study of a steel pedestrian bridge. We simulate realistic measurement conditions by incorporating aspects such as limited data and measurement noise when performing Bayesian model updating and damage detection. As this paper serves as a tutorial on Bayesian model updating with different MCMC methods, our primary focus is on numerically demonstrating Bayesian model updating across different structural types. In future studies, we plan to extend this work by applying Bayesian model updating to practical cases through experimental tests, as part of our ongoing research. Current Bayesian model updating relies heavily on the evaluation of the likelihood function. However, for complex models, such as multi-level models that embed sub-models, the likelihood function is often intractable and not available in a closed form. Likelihood-free inference (LFI) offers a viable alternative to traditional Bayesian model updating by bypassing the need for explicit likelihood calculations. Emerging techniques like Generative Adversarial Networks [85] and normalizing flows [86; 87] can further improve Bayesian model updating and structural damage detection by providing more accurate and computationally efficient approximations of the posterior distributions.