## 4 Framework for Reliability Analysis Incorporating Noisy Response Prediction
### Surrogate Reliability Analysis  
In conducting reliability analysis with the Kriging model, this study employs the simulation approach. Given that MCS may demand a substantial number of limit-state function evaluations to converge with an acceptable level of accuracy, especially when the probability of failure is small, subset simulation emerges as a viable option. Originating from the work of [PERSON] and [PERSON] [43; 44], this technique is crafted to mitigate the computational burden by addressing a sequence of simpler reliability problems with prescribed failure thresholds. In the subset approach, the failure domain is restructured as a decreasing sequence of failure domains as  
\[D_{f}=D_{f_{1}}\supset D_{f_{2}}\supset\cdots\supset D_{f_{m}},\ D_{f_{k}}=\{ \mathbf{x}:g(\mathbf{x})\leq\xi_{k}\} \tag{35}\]  
where \(\xi_{k}\) denotes a series of decreasing threshold values such that \(\xi_{1}>\xi_{2}>\cdots>\xi_{k}\) in relation to the limit-state function. The failure probability is then computed as a product of a sequence of conditional probabilities as  
\[P_{f}=P\left(\bigcap_{k=1}^{m}\mathbb{P}\Big{(}D_{f_{k}}\Big{)}\right)=\left( \prod_{i=1}^{m-1}\mathbb{P}(D_{f_{i+1}}|D_{f_{i}})\right)\cdot\mathbb{P}(D_{f _{1}}) \tag{36}\]  
The concept behind subset simulation is to estimate the failure probability by assessing these quantities. Estimating the failure probability using simulation approaches becomes more challenging as the failure probability decreases. In general, the smaller the probability of failure, the greater the number of simulation samples required to observe the failure event and calculate the failure probability. As discussed by [PERSON] and [PERSON] [43], appropriately assigning intermediate failure events allows the conditional probability in Equation (36) to be made sufficiently large, facilitating efficient evaluation through simulation approaches. In this study, a threshold value of 0.1 (\(\xi_{k}=0.1\)), as indicated in [43], is adopted to compute conditional probabilities.  
With a sample of size \(N_{\text{s}}\) of the input random vector \(\mathbf{X}\), MCS can be employed to compute \(\mathbb{P}(D_{f_{1}})\) in Equation (36) as  
\[\mathbb{P}(D_{f_{1}})\approx\frac{1}{N_{\text{s}}}\sum_{i=1}^{N_{\text{s}}}I_{ D_{1}}(\mathbf{x}_{i}^{\text{s}}) \tag{37}\]  
where \(\mathbf{x}_{i}^{\text{s}}\) denotes generated samples according to the PDF and \(I_{D_{1}}\) is the binary indicator function. The binary indicator function returns \"1\" if the limit-state function is negative or zero and \"0\" otherwise. To enhance the efficiency of sampling in the computation of the remaining conditional probability, this study employs the modified Metropolis method proposed by [PERSON] and [PERSON] [43], utilizing Markov chain simulation [45].  
The convergence is assessed based on the relative increment in the reliability index or the failure probability. The convergence criterion, expressed in terms of the change in reliability index, is formulated as  
\[\frac{\left|\hat{\beta}^{(k)}-\hat{\beta}^{(k-1)}\right|}{\hat{\beta}^{(k)}} \leq\epsilon_{\beta} \tag{38}\]  
where \(\hat{\beta}^{(k)}\) represents the estimated reliability index at the \(k\)-th iteration and \(\epsilon_{\hat{\beta}}\) denotes a prescribed tolerance value.