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In that case observations $y(\mathbf{x})$ are generated by Gaussian Process with zero mean and covariance function $\mathrm{Cov}\left(y(\mathbf{x}), y(\mathbf{x}^\prime)\right) = k(\mathbf{x}, \mathbf{x}^\prime) + \tilde{\sigma}^2\delta(\mathbf{x}- \mathbf{x}^\prime)$, where $\delta(\mathbf{x})$ is a Dirac delta funciton.  Thus, aposterior mean funciton of Gaussian Process $f(\mathbf{x})$ in the points of test set $X_*$ takes form:  \begin{equation}%\label{meanNoise} \begin{equation}  \label{meanNoise}  \hat{f}(X_*) = K_* \left(K + \tilde{\sigma}^2 I\right)^{-1} Y,  \end{equation}  where $I$ -- identity matrix of size $(N \times N)$.