## 3 Markov Chain Monte Carlo Algorithms
### Transitional Markov Chain Monte Carlo (TMCMC)
#### 3.2.3 Implementation and Algorithmic Steps
of Markov chains; \(d\): parameter dimensions; \(M\): the number of samples \(p(\boldsymbol{\theta})\): prior distribution For\(i\) = 1,2,\(\cdots\), \(M\): Draw the candidate samples \(\boldsymbol{\theta}_{i}\)\(\sim\)\(p(\boldsymbol{\theta})\) for each chain End For\(j\) = 1,2,\(\cdots\), \(N\): Compute evolution of each chain and current sample candidates using Equation (24)  Modify component of sample candidates with probability \(1-\widehat{\mathcal{CH}}\) using Equation (25)  Compute acceptance rate based on available samples  Determine if accept or reject samples using Equation (23)  Remove potential aberrant chains using interquartile range statistics in Equation (26)  Check convergence status using Equation (27) If\(R_{stat}\) > 1.2: Repeat computation from Equations (24) to (27) Else: End  Save samples from all chains and estimate posterior statistics
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**Algorithm 3****DREAM [50]**

It is important to note that the selection of the threshold value of 1.2 for the \(R_{stat}\) is based on empirical studies [25; 50]. However, determining an appropriate threshold is inherently challenging and may need to be adjusted based on the specific problem at hand. For instance, thresholds of 1.1 or even 1.05 may be more suitable in certain contexts [52]. [PERSON] et al. [52] also suggest that \(R_{stat}\) values below 1.1 are generally acceptable for most applications. Additionally, thresholds can be adaptively adjusted using termination criteria that consider the relationship between the effective sample size (ESS) and the Gelman-Rubin statistic [53]. Discussions on selecting an appropriate \(R_{stat}\) threshold can be found in [54; 55]. It is crucial to recognize that while \(R_{stat}\) is a valuable tool for assessing MCMC convergence, it is not the sole diagnostic measure. Comprehensive diagnostic analysis should include various criteria, such as trace plot visualization, ESS for mixing diagnostics [56; 57], and sampler precision [58]. For a more detailed exploration of diagnostic criteria, refer to [PERSON] et al. [52]. **4. Comparative Studies on Numerical Examples**

In this Section, the three MCMC algorithms described in Section 3 are applied to Bayesian model updating for structure damage detection. Three engineering structures, namely, a forty-story shear building, a two-span continuous concrete beam, and a pedestrian bridge, are investigated with increasing structural complexity in terms of the number of DOFs. The damage scenarios are intentionally introduced. In all case studies, the performance of each MCMC is evaluated by the accuracy of the damage detection, parameter estimation, and computational cost. It is important to note that the damage scenarios in the numerical examples are intended to represent relative changes in the elastic modulus. For instance, in Tables 1-3, a value of \(-20\%\) corresponds to \((E_{d}-E_{u})/E_{u}\), where \(E_{d}\) and \(E_{u}\) represent the elastic modulus in the damaged and undamaged conditions, respectively. The decrease in elastic modulus is a common representation of stiffness reduction, as stiffness is directly related to the elastic modulus [59, 60, 24, 61]. Additionally, we define the stiffness parameter (updating parameter) \(\theta\) as the ratio of the true elastic modulus (damaged condition) to the nominal elastic modulus (healthy condition). A \(\theta\) value of unity indicates an undamaged condition. The values used in our damage scenarios are based on findings from the literature on Bayesian model updating and damage detection in various structures, including laboratory-scaled shear buildings [31, 62], steel truss bridges [37], offshore platforms [63], and steel frames [64, 65]. Furthermore, the structures analyzed in this study are all based on numerical models. Since the focus of the paper is not on the specific material properties of the structures, we did not specify the material types in detail.