## 1 Introduction
[PERSON] and [PERSON] [21] proposed an automated damage detection framework by integrating surrogate model-based BMUA with DRAM for a historic tower. [PERSON] et al. [22] also combined the response surface model with DRAM for nonlinear model updating in a three-story frame building. TMCMC, on the other hand, adopts a series of transitional distributions, iteratively transitioning from the prior to the posterior by defining a tempering parameter. This approach makes posterior inference more efficient and capable of sampling from complex posterior distributions. The feasibility of TMCMC in structural damage detection has also been explored. [PERSON] et al. [23] proposed a probabilistic damage detection framework using Ultrasonic Guided Waves and TMCMC for complex structures. [PERSON] et al. [24] applied vibration-based BMUA and TMCMC for incremental damage detection on a real-world steel truss bridge. In contrast, multi-chain MCMC estimates the posterior distribution by running multiple trajectories or Markov chains in parallel. This approach theoretically improves the quality of posterior samples by integrating information from different inde pendent chains. The interaction and mixing of information from these chains enhance the exploration of the full parameter space, allowing for the simultaneous search for multiple solutions. The use of multi-chain MCMC offers several advantages, particularly when dealing with complex posterior distributions characterized by sharp peaks, local optima, and long tails. [PERSON] [25] provided an exhaustive overview of multi-chain MCMC and its application in the wide interdisciplinary field. The studies in [25] also illustrate that single-chain methods may be locally trapped, and their performance may degrade with problem complexity and the number of model parameters. The multi-chain methods exhibit desirable performance in high-dimensional problems due to collective power in sampling. Popular multi-chain MCMCs include the Shuffled Complex Evolution Metropolis (SCEM) algorithm [26], the Differential Evolution Markov Chain (DE-MC) [27], the population MCMC [28], the Differential Evolution Adaptive Metropolis (DREAM) [29], etc. However, only a few works have explored the capability of multi-chain MCMC in structural damage detection for SHM. [PERSON] and [PERSON] [30] applied multiple parallel and adaptive Markov chains and Bayesian analysis to identify the stiffness reduction for a plate structure. [PERSON] and [PERSON] [31] proposed a new BMUA with DREAM to perform damage detection for a three-story shear building. [PERSON] et al. [32] presented damage identification in a cracked plate using population MCMC and Bayes rule. A longstanding problem in BMU with MCMC is the curse of dimensionality. According to the literature review [31; 33; 34], most MCMC methods perform well with only a few uncertain parameters. Attempts to update many parameters often compromise accuracy [15]. Consequently, BMU for structural damage detection is typically performed in low-dimensional space through numerical, laboratory, and field tests. While a common strategy in damage detection is to assign a single parameter to different components or member groups, considering more uncertain parameters and finer parameterization is desirable. This approach helps inspectors narrow down detection areas and achieve element-level damage detection. Another challenge is that MCMC performance is application-specific, requiring users to select the appropriate MCMC method carefully and understand its mechanism. However, to the best of the authors' knowledge, there is no comprehensive work providing in-depth instruction on BMU with various MCMC methods applied to structural damage detection in SHM. This paper presents a comparative study of three advanced MCMC techniques for Bayesian model updating in structural damage detection. The three MCMC algorithms studied are DRAM, TMCMC, and DREAM. Detailed introductions, mechanisms, and guidelines for each algorithm are provided.