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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{eq:mean_noise} \begin{equation}\label{eq:mean-noise}  \hat{f}(X_*) = K_* \left(K + {\sigma}^2 I \right)^{-1} Y,  \end{equation}  where $I$ -- identity matrix of size $(N \times N)$.  Note, that noise variance $\tilde{\sigma}^2$ in (\ref{eq:mean_noise}) (\ref{eq:mean-noise})  in fact leads to regularization and more generalization ability of the resulting regression. Wherein the aposteriori covariance function of Gaussian Process in the points of test set takes form: \begin{equation}\label{eq:covariancenoise}\mathrm{V} \begin{equation}\label{eq:covariance-noise}  \mathrm{V}  \left[X_*\right] = K(X_*, X_*) + \tilde{\sigma}^2 I_* - K_* \left(K + \tilde{\sigma}^2 I \right)^{-1} K_*^T, \end{equation}  where $K(X_*, X_*) = \left[k(\mathbf{x}_i, \mathbf{x}_j) \middle| i, j = 1, \dots, N_*\right]$ and $I_*$ -- identity matrix of size $(N_* \times N_*)$.