## 3 Bayesian Updating with Detection Information
### Updating of the Reinforcement Initial Corrosion Model  
The updated chloride diffusion model calculates the limit state of initial corrosion durability at structure detection time \(t_{0}\):  
\[G=C_{cr}-C(x_{d},t_{0})=C_{cr}-\epsilon_{1}\cdot C_{s}\Big{[}1-erf\Big{(}x_{ d}/\Big{(}2\sqrt{D_{Cl}t_{0}}\Big{)}\Big{)}\Big{]} \tag{13}\]  
According to the limit state equation of initial corrosion durability of steel bars, considering the probability distribution of concrete cover thickness and the prior critical chloride ion concentration model, the prior corrosion probability of reinforcement is calculated as follows:  
\[ \begin{array}{l}p_{\text{cor}}(t_{0})=\Pr\{G<0\}=\Pr\{C_{cr}-\epsilon_{1} \cdot C_{s}\big{[}1-erf\big{(}x_{d}/\big{(}2\sqrt{D_{Cl}t_{0}}\big{)}\big{)} \big{]}<0\}\\ 0<p_{\text{cor}}(t_{0})<1\end{array}  \tag{14}\]  
The probabilistic model for critical chloride concentration is commonly used to assess the risk of reinforcement corrosion in concrete structures, taking into account the uncertainty and variability of chloride ion concentration. The critical chloride ion concentration values can vary depending on several factors, including the type of cement, the quality of concrete, the type of reinforcement, and the environmental conditions. [PERSON] [31] suggested that the value of \(C_{\sigma}\) followed the normal distribution mean of 2.4 kg/m\({}^{3}\) and the COV (coefficient of variation) of 0.2. [PERSON] et al. [32] advised \(C_{cr}\) as the mean of 3.07 kg/m\({}^{3}\) and a COV (coefficient of variation) of 0.41. The Japanese Civil Society criteria [33] considered \(C_{cr}\) with 0.3-2.4 kg/m\({}^{3}\) when predicting the service life. For reinforced concrete with high durability requirements, the chloride ion mass of \(C_{cr}\) does not exceed 0.3 kg/m\({}^{3}\). It is a higher value of \(C_{cr}\) than in laboratory conditions measured in the actual environment, with 1.2-2.4 kg/m\({}^{3}\). Bayesian updating is a statistical method based on Bayesian theory, used to update existing probability distributions or model parameters based on new evidence or data. Bayesian updating describes how to update the conditional probability of an event based on new evidence. The theorem can be expressed as follows:  
\[P(\text{A}\ket{\text{B}})=\frac{P(\text{A})P(\text{B}\ket{\text{A}})}{P(\text {B})} \tag{15}\]  
where \(P(\text{A}\ket{\text{B}})\) is the probability of event A occurring given that event B has occurred, \(P(\text{B}\ket{\text{A}})\) is the probability of event B occurring given that event A has occurred, \(P(\text{A})\) is the prior probability of event A occurring, and \(P(\text{B})\) is the total probability of event B occurring. In Bayesian updating, the prior distribution represents the estimation of the probability distribution of model parameters before obtaining new data, while the posterior distribution is the updated probability distribution of model parameters after considering the new data. The goal of Bayesian updating is to obtain a more accurate posterior distribution by combining the prior distribution with new data. The likelihood function is a key concept that represents the probability of observing new data given the model parameters. In Bayesian updating, the likelihood function is used to measure the fit between the new data and the model parameters. Assuming that there are \(n\) samples of detected half-cell potential, \(m\) are corrosion samples. It is expressed as event \(H\) here. Then, the likelihood function of the rust probability can be expressed as follows:  
\[l(p_{\text{cor}}|H)=\binom{n}{m}p_{\text{cor}}^{m}(1-p_{\text{cor}})^{n-m} \tag{16}\]  
where \(\binom{n}{m}\) refers to the number of combinations of \(m\) samples selected from \(n\) samples.