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Note, that noise variance $\tilde{\sigma}^2$ in (\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}  \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:covariance-noise} \label{eq:covariancenoise}  \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 $(N_* \times N_*)$.