## 3 Bayesian Updating with Detection Information
### Updating of the Reinforcement Initial Corrosion Model
According to Bayesian update theory, the posterior probability density function of corrosion probability is the following:  
\[f^{\prime\prime}(p_{\text{cor}}|H)=\frac{l(p_{\text{cor}}|H)f^{\prime}(p_{ \text{cor}})}{\int\limits_{p_{\text{cor}}}l(p_{\text{cor}}|H)f^{\prime}(p_{ \text{cor}})d{p_{\text{cor}}}}=\frac{p_{\text{cor}}^{m}(1-p_{\text{cor}})^{n- m}f^{\prime}(p_{\text{cor}})}{\int\limits_{p_{\text{cor}}}p_{\text{cor}} ^{m}(1-p_{\text{cor}})^{n-m}f^{\prime}(p_{\text{cor}})d{p_{\text{cor}}}} \tag{17}\]  
where \(f^{\prime}(p_{\text{cor}})\) is the prior probability density function of the corrosion probability, which can be assumed to follow a normal distribution (truncation), Weibull distribution, or Beta distribution. The mean value \(\mu^{\prime}_{p_{\text{cor}}}\) was calculated using Formula (14), and a coefficient of variation of 0.5 was assumed. Experiments have shown that the selection of prior distributions has little impact on the updated results. Considering that the Beta distribution has the characteristics of a conjugate prior distribution, it is recommended to use the Beta distribution for convenient calculation. In the case of the test result \(H\), the mean value of the posterior probability density function of the corrosion probability \(f^{\prime\prime}(p_{\text{cor}}|H)\) was \(\mu^{\prime\prime}_{p_{\text{cor}}}\). Using the posterior mean of the density function of corrosion probability, with the updatedchloride diffusion model and the measured concrete cover thickness, the updated mean value (assuming constant of standard deviation) of \(C_{cr}\) was calculated as follows:  
\[\mu^{\prime\prime}_{C_{cr}}=\left\{\mu^{\prime\prime}_{C_{cr}}:\Pr[C_{cr}-C(x_{d },t_{0})<0]=\mu^{\prime\prime}_{\text{Pcor}}\right\} \tag{18}\]  
The above formula adopts the traversal algorithm, selects the mean of \(C_{cr}\) in the effective interval, adopts the Monte Carlo simulation algorithm, and calculates the initial corrosion probability \(p_{\text{cor}}(t_{0})\) of the steel bar at the time point \(t_{0}\). The mean of \(C_{cr}\) consistent with the updated mean of the rust probability was selected and determined in the above equation \(\mu^{\prime\prime}_{C_{cr}}\) to complete the right updating of \(C_{cr}\)'. With the updating \(C_{cr}\)', combined with the updated model of chloride ion diffusion and protective layer thickness probability distribution, Formula (19) was used to predict the initial corrosion probability of \(t\) (\(t>t_{0}\)) years, the prediction results of the initial corrosion probability of future reinforcement with comprehensive theoretical knowledge and historical information of components. The updated method above essentially assumes that the corrosion probability of different components in the same structure is the same in a certain year. Subsequently, the updated initial corrosion time \(t_{ini}\) was calculated, providing the basis for the updating of the rust swelling and cracking model:  
\[t_{ini}={x_{d}}^{2}\cdot(4D_{CI})^{-1}\cdot\left[erf^{-1}(1-(C_{cr})/\varepsilon _{1}C_{s})\right]^{-2} \tag{19}\]  
In the formula, an updated distribution model was used for each parameter, and the calculated initial corrosion time is a variable rather than a specific value. In summary, the steps for updating the reinforcement initial corrosion model are as follows:  
1. Calculate the mean value of the prior corrosion probability of reinforcement \(p_{\text{cor}}(t_{0})\) using Formula (14) with the updated chloride diffusion model in Section 3.1. List the mean value of the prior probability \(\mu^{\prime}_{P_{\text{cor}}(t_{0})}\) and the coefficient of variation as 0.5. 2. In the case of the test result \(H\) (number of corrosion samples in a number of test samples), calculate the mean value and the standard deviation of the posterior probability density function of the corrosion probability \(f^{\prime\prime}(p_{\text{cor}}|H)\) as \(\mu^{\prime\prime}_{P_{\text{cor}}(t_{0})}\) and \(\sigma^{\prime\prime}_{P_{\text{cor}}(t_{0})}\) using Formula (17). 3.