## 2 Statistical Inverse Theory  
The analysis of a physical system is generally conducted according to the following steps: (1) parameterizing the system; (2) establishing the forward model; and (3) establishing the inverse model [19]. When evaluating a bridge, stress can be estimated by means of the forward model connecting the structural information, load information, and stress. Since the monitored strain was derived from the stress action, the inverse model should be built to utilize the monitored strain information. In the establishment of the inverse model, the model space, which is derived from selected parameters and the forward model, and the data space, which is derived from the monitored results, are needed. The selection of the model parameters is not unique. A selection represents a kind of system parameterization. A series of physical tests needs to be carried out to determine the boundary conditions of the model space, i.e., the variation range of the model parameters. Generally, the quantile values with a certain confidence level for the prior distribution of all the model parameters are considered to be in the variation range of the parameters. For the physical system, the data space of the system is composed of the monitored results. Since the parameters in the data space have a direct causal relationship with the model parameters in the model space [20], the evaluation model between the two series of parameters can be established and one or multiple solutions can be derived for the model.  
Statistical inverse analysis essentially randomizes the deterministic problem based on the Bayesian statistical method. In statistical inverse analysis, the input and output are not certain values, but rather random variables and their statistical characteristics. Based on the Bayesian Theory, when the data space is determined, the statistical characteristics of the model space are the conditional probability of the statistical parameters:  
\[p(\mathbf{m}\,|\mathbf{d}\,)=\frac{p(\mathbf{d}\,|\mathbf{m}\,)p(\mathbf{m})} {p(\mathbf{d})}, \tag{1}\]  
where \(\mathbf{m}\) is the model parameter vector and \(\mathbf{d}\) is the monitored parameter vector.  
Since \(p(\mathbf{d})\) can be considered constant, Equation (1) can be expressed as  
\[p(\mathbf{m}\,|\mathbf{d})\propto L(\mathbf{m})p(\mathbf{m}), \tag{2}\]  
where \(L(\mathbf{m})=p(\mathbf{d}/\mathbf{m})\) is the likelihood function, as shown in Equation (3):  
\[L(\mathbf{m})=(\pi v)^{-\frac{N}{2}}\exp\Biggl{\{}-\frac{|\mathbf{d}-D(\mathbf{ m})|^{2}}{v}\Biggr{\}}, \tag{3}\]  
where \(N\) is the number of the parameters in the data space, \(D(\mathbf{m})\) is the result of the model parameter from the forward model, and \(v\) is the corresponding variance of \(D(\mathbf{m})\).  
By normalizing the right partition of Equation (2), the expression for the conditional probability can be derived as  
\[p(\mathbf{m}\,|\mathbf{d})=\frac{L(\mathbf{m})p(\mathbf{m})}{\int L(\mathbf{m }^{\prime})p(\mathbf{m}^{\prime})d\mathbf{m}^{\prime}}. \tag{4}\]  
In order to obtain the maximum likelihood probability of the model parameters, the square sums of the residual between the measurement and prediction are calculated as cost functions, as shown in Equation (5):  
\[\Phi(\mathbf{m})=|\mathbf{d}-D(\mathbf{m})|^{2}. \tag{5}\]  
The objective of inverse analysis is to determine the series of parameters under global optimization and their probability when the corresponding cost functions approach a minimum, and then modify the prior distribution of the model parameters.