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Thus, aposterior mean funciton of Gaussian Process $f(\mathbf{x})$ in the points of test set $X_*$ takes form:  \begin{equation}  \hat{f}(X_*) = K_* \left(K + {\sigma}^2 I \right)^{-1} Y,  \label{eq:mean-noise} \label{eq:meannoise}  \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:meannoise})  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}  \mathrm{V} \left[X_*\right] = K(X_*, X_*) + \tilde{\sigma}^2 I_* - K_* \left(K + \tilde{\sigma}^2 I \right)^{-1} K_*^T,  \label{eq:covariancenoise}