## 2 Background  
### Bayesian Model Updating  
Bayesian model updating (BMU) is established based on [PERSON] theorem that describes the probability of event A given event B [35]. BMU has widespread applications and its popular use in the engineering community is structural damage detection for SHM [36; 37; 31; 38; 39]. BMU profits in updating the prior information of parameters of interest using available observations, resulting in the posterior distribution of these parameters. Assume a set of to-be-updated model parameters \(\Theta=\left(\theta^{(1)},\theta^{(2)},\cdots,\theta^{(N_{\theta})}\right)\), where \(N_{\theta}\) is the number of model parameters, and we also have a set of M-independent and identically distributed (i.i.d.) measured data \(\mathbf{D}=(D_{1},D_{2},\cdots,D_{M})\). Following [PERSON] theorem, the posterior distribution \(p(\Theta|\mathbf{D})\) of model parameters is expressed as  
\[p(\theta|\mathbf{D})=\frac{p(\mathbf{D}|\Theta)p(\theta)}{p(\mathbf{D})}, \tag{1}\]  
where \(p(\theta)\) denotes the prior distribution describing existing knowledge, engineers' opinions, and expected physical meaning, and \(p(\mathbf{D}|\theta)\) denotes the likelihood function reflecting the degree of belief in how well a model explains the actual observation with the parameter vector \(\theta\). The likelihood function can also be viewed as the probability function of observing data given certain parameters. However, throughout the paper, we use the term likelihood function in accordance with [PERSON] theorem for clarity and consistency. \(p(\mathbf{D})\) denotes the normalizing constant given by \(\int p(\mathbf{D}|\Theta)p(\theta)d\theta\) to ensure \(\int p(\theta|\mathbf{D})d\theta=1\). This constant also refers to the evidence that plays a paramount role in Bayesian model selection and averaging [40]. Without directly solving the integral to obtain \(p(\mathbf{D})\), Equation (1) can be rewritten as  
\[p(\theta|\mathbf{D})\ \propto\ p(\mathbf{D}|\Theta)p(\theta), \tag{2}\]  
where \"\(\propto\)\" stands for \"proportional to\". The prior distribution \(p(\theta)\) represents the initial knowledge of model parameters before any measurements are available. In most cases, the prior is decided by the engineers' judgment [41]. In structural damage detection, the model parameters usually concern stiffness parameters, assuming that the mass is known. The allowable variation in stiffness can be taken as \(\pm 20\%\) or \(\pm 30\%\) according to general engineering judgement [42; 43]. Beyond this, a structure tends to be completely out of service or may even collapse. In this study, observations are modal data, including natural frequencies and modal shapes. Structural damage is closely correlated with variations in modal parameters that can be identified from operation modal analysis (OMA), which is also called output-only or natural-excitation modal analysis [44]. Assume observation \(\mathbf{D}\) contains \(N_{m}\) sets of modal parameters identified from \(N_{k}\) sets of dynamic measurements. We also assume the modal parameters are i.i.d. from mode to mode. The likelihood function has the following mathematical form. \[p(\mathbf{D}|\mathbf{\theta})=\prod_{j=1}^{N_{k}}\prod_{i=1}^{N_{m}}p\bigg{(} \widetilde{f}_{i,j}\bigg{|}\mathbf{\theta}\bigg{)}p\bigg{(}\widetilde{\mathbf{ \Phi}}_{i,j}\bigg{|}\mathbf{\theta}\bigg{)}, \tag{3}\]  
where \(\widetilde{f}_{i,j}\) and \(\widetilde{\mathbf{\Phi}}_{i,j}\) are, respectively, identified as natural frequencies and mode shapes; \(i=1,2,\cdots,N_{m}\), \(j=1,2,\cdots,N_{k}\), \(N_{m}\) is the number of modes; and \(N_{k}\) is the number of tests.