## 2 Background
### Bayesian Model Updating
The error functions between the \(i\)-th-measured frequency and mode shape and the model-derived counterparts are defined as  
\[e_{f_{i,j}}=\widetilde{f}_{i,j}-f_{i,j}(\mathbf{\theta}), \tag{4}\]  
\[e_{\mathbf{\Phi}_{i,j}}=\widetilde{\mathbf{\Phi}}_{i,j}-\mathbf{\Phi}_{i,j}( \mathbf{\theta}), \tag{5}\]  
where \(f_{i,j}(\mathbf{\theta})\) and \(\mathbf{\Phi}_{i,j}(\mathbf{\theta})\) are, respectively, the analytical natural frequency and the mode shape given the model parameter \(\mathbf{\theta}\), \(i=1,2,\cdots,N_{m}\), \(j=1,2,\cdots,N_{k}\). Assume the error function \(e\) between the observation and model output follows zero-mean Gaussian distributions with a fixed variance. Consequently, the likelihood function of natural frequency given \(\mathbf{\theta}\) is written as  
\[p\bigg{(}\widetilde{f}_{i,j}\bigg{|}\mathbf{\theta}\bigg{)}=\Big{(}2\pi \sigma_{f_{i}}^{2}\Big{)}^{-\frac{1}{2}}\text{exp}\Bigg{\{}-\frac{1}{2}\frac{ \left(\widetilde{f}_{i,j}-f_{i,j}(\mathbf{\theta})\right)^{2}}{\sigma_{f_{i} }^{2}}\Bigg{\}}, \tag{6}\]  
where \(\sigma_{f_{i}}^{2}\) is the variance of the \(i\)-th-measured natural frequency that equals the sampling variance in the \(N_{k}\) dynamic tests in this study. Similarly, the likelihood function of mode shape given \(\mathbf{\theta}\) is written as  
\[p\bigg{(}\widetilde{\mathbf{\Phi}}_{i,j}\bigg{|}\mathbf{\theta}\bigg{)}=\big{(} 2\pi\Sigma_{\mathbf{\Phi}_{i}}\big{)}^{-\frac{1}{2}}exp\Bigg{\{}-\frac{1}{2} \bigg{(}\widetilde{\mathbf{\Phi}}_{i,j}-\mathbf{\Phi}_{i,j}(\mathbf{\theta}) \bigg{)}^{T}\Sigma_{\mathbf{\Phi}_{i}}^{-1}\bigg{(}\widetilde{\mathbf{\Phi}}_ {i,j}-\mathbf{\Phi}_{i,j}(\mathbf{\theta})\bigg{)}\Bigg{\}}, \tag{7}\]  
where \(\Sigma_{\mathbf{\Phi}_{i}}\) is a diagonal covariance matrix of the \(i\)-th-measured mode shape, whose diagonal elements are equal to the variances of the mode shape in the \(N_{k}\) dynamic tests. Inserting Equations (6) and (7) into Equation (3) yields the following likelihood function:  
\[p(\mathbf{D}|\mathbf{\theta})=c_{0}\prod_{j=1}^{N_{k}}\prod_{i=1}^{N_{m}}\text {exp}\bigg{(}-\frac{1}{2}J(\mathbf{\theta};\mathbf{D})\bigg{)}, \tag{8}\]  
\[J(\mathbf{\theta};\mathbf{D})=\Bigg{\{}\frac{\left(\widetilde{f}_{i,j}-f_{i,j }(\mathbf{\theta})\right)^{2}}{\sigma_{f_{i}}^{2}}\Bigg{\}}+\Bigg{\{}\frac{1}{ \Sigma_{\mathbf{\Phi}_{i}}}\bigg{(}\widetilde{\mathbf{\Phi}}_{i,j}-\mathbf{ \Phi}_{i,j}(\mathbf{\theta})\bigg{)}^{T}\bigg{(}\widetilde{\mathbf{\Phi}}_{i,j}-\mathbf{\Phi}_{i,j}(\mathbf{\theta})\bigg{)}\Bigg{\}}, \tag{9}\]  
where \(c_{0}\) includes all constants in Equations (6) and (7). It should be pointed out that it would be more convenient to use the logarithmic of the likelihood function, which refers to the log-likelihood given by  
\[\text{log}(\mathbf{D}|\mathbf{\theta})=c_{0}\sum_{j=1}^{N_{k}}\sum_{i=1}^{N_ {m}}-\frac{1}{2}J(\mathbf{\theta};\mathbf{D}). \tag{10}\]  
The log-likelihood function in Equation (10) facilitates the numerical calculation of the likelihood function. Since the likelihood function requires the product computation of each measurement and each mode, as shown in Equation (8), various advanced MCMC methodshave been developed for Bayesian model updating. In Section 3, single-chain MCMC, e.g., DRAM and TMCMC, and multi-chain MCMC, e.g., DREAM, will be reviewed.