## 1 Introduction  
Finite element model updating (FEMU) is a popular damage detection method for civil infrastructures within structural health monitoring (SHM). FEMU minimizes discrepancies between model-predicted responses and measured data by adjusting structural model parameters [1,2]. In a parameterized FE model in an undamaged condition, any damage occurrence will lead to alterations in structural characteristics. Such alterations can be measured by the deviation between structural damage and FE model parameters. FEMU aims to minimize these deviations, aligning model parameters with those of the actual damaged structure, and enabling damage detection, localization, and quantification [3,4]. The traditional FEMU approach is deterministic, searching a point-estimate or unique solution of the model parameter, such as direct FEMU [5,6], sensitivity-based FEMU [7,8], and evolutionary algorithm-based FEMU [9,10], etc. However, these methods often neglect uncertainties arising from limited sensors, noise in measured data, environmental variability, operational loading, and modeling errors. These factors inevitably induce uncertainties in structural damage detection [11]. FEMU with uncertainty consideration for damage detection can be categorized into probabilistic and non-probabilistic approaches. Probabilistic approaches, especially the Bayesian model updating approach (BMUA), have extensive applications for damage detection in SHM. [PERSON] and co-workers [12] are pioneers in establishing the fundamental framework of BMUA. Based on the Bayesian point of view, the prior information of parameters is in conjunction with the likelihood function, resulting in the posterior distributions of model parameters. BMUA updates model parameters as probability density functions (PDFs), naturally quantifying uncertainties [13]. This allows for robust and accurate damage detection decisions. Nevertheless, BMUA faces challenges with analytically intractable posterior PDFs involving multidimensional integrals [12]. To address this difficulty, Markov Chain Monte Carlo (MCMC) is widely used to approximate the posterior PDF of model parameters. MCMC avoids evaluating multidimensional integrals by iteratively drawing samples from the proposal distribution. Accepted samples, based on the Metropolis-Hastings (MH) rule, form a Markov chain whose converged state distribution is the target posterior PDF [14]. MCMC extends the application of BMUA for structural damage detection by flexibly and powerfully inferring posteriors. However, the success of MCMC depends on the problem dimension and sampling mechanism. The former affects the sample acceptance rate and the ability to explore high-probability regions. The latter determines the efficiency of reaching a stationary state of the Markov chain. An improper sampling mechanism can lead to incorrect posterior inference [15]. MCMC algorithms can be broadly categorized into single-chain and multi-chain methods based on the number of Markov chains used to infer posteriors. Single-chain MCMC utilizes one Markov chain to generate posterior samples. The earliest and most classical single-chain MCMC methods are the Metropolis-Hastings (MH) algorithm and its variant, the Gibbs sampler. However, MH has a slow convergence rate and is inefficient for high-dimensional problems [16], while the Gibbs sampler struggles with nonstandard conditional distributions [17]. Two advanced single-chain MCMC algorithms are the Delayed Rejection Adaptive Metropolis (DRAM) [18] and the transitional MCMC (TMCMC) [19]. DRAM improves MH by combining the delayed rejection (DR) algorithm and the adaptive metropolis (AM) algorithm, accelerating convergence and enhancing the acceptance rate. DR adapts proposals rather than maintaining the same sample on rejection, while AM adjusts the covariance of the proposal distribution over time. In structural damage detection, [PERSON] et al. [20] applied sparse Bayesian learning with variational inference and DRAM for damage detection in a laboratory-scale steel frame.