## 4 Framework for Reliability Analysis Incorporating Noisy Response Prediction
### Estimates of the Hyperparameters with Noise  
The optimization problem of maximum likelihood estimation in Equation (17) without noise can be reformulated as an equivalent minimization problem in natural logarithm form, expressed as:  
\[\min_{\Theta,\mathbf{m},\sigma^{2}}\Bigl{(}-\text{log}\mathcal{L}(\mathbf{m}, \sigma^{2},\Theta;\boldsymbol{\mathcal{Y}}_{D})\Bigr{)} \tag{23}\]  
Based on the first-order optimality conditions for unconstrained optimization, the following derivations can be obtained:  
\[ \begin{split}\
abla_{\mathbf{m}}\log\mathcal{L}(\mathbf{m}, \sigma^{2},\Theta;\boldsymbol{\mathcal{Y}}_{D})&=\frac{1}{2 \sigma^{2}}(\boldsymbol{\mathcal{F}}^{T}\boldsymbol{\kappa}^{-1}\boldsymbol{ \mathcal{Y}}_{D}-\boldsymbol{\mathcal{F}}^{T}\boldsymbol{\kappa}^{-1} \boldsymbol{\mathcal{F}}\mathbf{m})=\mathbf{0}\\ \frac{\partial\log\mathcal{L}(\mathbf{m},\sigma^{2},\Theta; \boldsymbol{\mathcal{Y}}_{D})}{\partial\sigma^{2}}&=-\frac{n}{2} +\frac{1}{2\sigma^{2}}(\boldsymbol{\mathcal{Y}}_{D}-\boldsymbol{\mathcal{F}} \mathbf{m})^{T}\boldsymbol{\kappa}^{-1}(\boldsymbol{\mathcal{Y}}_{D}- \boldsymbol{\mathcal{F}}\mathbf{m})=0\end{split}  \tag{24}\]  
From Equation (24), the generalized least-squares estimates, \(\widetilde{\mathbf{m}}\) and \(\widetilde{\sigma}^{2}\), of \(\mathbf{m}\) and \(\sigma^{2}\) are obtained as  
\[ \begin{split}\widetilde{\mathbf{m}}&=\mathbf{m}( \Theta)=(\boldsymbol{\mathcal{F}}^{T}\boldsymbol{\kappa}^{-1}\boldsymbol{ \mathcal{F}})^{-1}(\boldsymbol{\mathcal{F}}^{T}\boldsymbol{\kappa}^{-1} \boldsymbol{\mathcal{Y}}_{D})\\ \widetilde{\sigma}^{2}&=\sigma^{2}(\Theta)=\frac{1} {n}(\boldsymbol{\mathcal{Y}}_{D}-\boldsymbol{\mathcal{F}}\widetilde{\mathbf{m} })^{T}\boldsymbol{\kappa}^{-1}(\boldsymbol{\mathcal{Y}}_{D}-\boldsymbol{ \mathcal{F}}\widetilde{\mathbf{m}})\end{split}  \tag{25}\]  
With the generalized least-square estimate, the optimization problem in Equation (23) can be simplified as  
\[\min_{\Theta}-\text{log}\mathcal{L}(\Theta;\boldsymbol{\mathcal{Y}}_{D}) \tag{26}\]  
When the noise is homoscedastic, a constant and uniform variance for all observed responses is indicated, and when it is heteroscedastic, independent noise with varying variances for each observed response without correlation is signified. In the presence of homoscedastic or heteroscedastic noise with a known variance, the maximum likelihood function can be expressed using the PDF of the multivariate normal distribution as follows:  
\[\mathcal{L}(\mathbf{m},\sigma^{2},\sigma_{*}^{2},\Theta;\boldsymbol{\mathcal{ Y}}_{D})=\frac{det(\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-0.5}}{(2\pi)^{0 5n}}\cdot\exp\Bigl{(}-0.5(\boldsymbol{\mathcal{Y}}_{D}-\boldsymbol{\mathcal{F }}\mathbf{m})^{T}(\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-1}(\boldsymbol{ \mathcal{Y}}_{D}-\boldsymbol{\mathcal{F}}\mathbf{m})\Bigr{)} \tag{27}\]  
The generalized least-square estimate of \(\mathbf{m}\) is a function dependent on \(\Theta,\sigma^{2},\sigma_{*}^{2}\). Employing the first optimality condition, the subsequent expressions are derived:  
\[\
abla_{\mathbf{m}}\log\mathcal{L}(\mathbf{m},\sigma^{2},\sigma_{*}^{2}, \Theta;\boldsymbol{\mathcal{Y}}_{D})=\frac{1}{2}(\boldsymbol{\mathcal{F}}^{T} (\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-1}\boldsymbol{\mathcal{Y}}_{D}- \boldsymbol{\mathcal{F}}^{T}(\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-1} \boldsymbol{\mathcal{F}}\mathbf{m})=\mathbf{0} \tag{28}\]  
Thus, the generalized least square estimate \(\widetilde{\mathbf{m}}\) is computed by  
\[\widetilde{\mathbf{m}}=\mathbf{m}(\Theta)=(\boldsymbol{\mathcal{F}}^{T}( \sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-1}\boldsymbol{\mathcal{F}})^{-1}( \boldsymbol{\mathcal{F}}^{T}(\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*})^{-1} \boldsymbol{\mathcal{Y}}_{D}) \tag{29}\]  
In contrast to the noise-free scenario, the variance estimate \(\widetilde{\sigma}^{2}\) cannot be analytically derived because of the implicit terms \(\sigma^{2}\boldsymbol{\kappa}+\Sigma^{*}\). It needs to be determined by solving the optimization problem.