deletions | additions       diff --git a/GP gradients1.tex b/GP gradients1.tex  new file mode 100644  index 0000000..9b114c1  --- /dev/null  +++ b/GP gradients1.tex 
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Moreover, knowing mena and covariance funcitons one can have an aposteriori estimate of mean and variance of the Gaussian Process gradient in the points of test set.   \begin{lemma}  Given two line segments whose lengths are $a$ and $b$ respectively there   is a real number $r$ such that $b=ra$.  \end{lemma}  Indeed, if  \[  \mathbf{g}(\mathbf{x}_0) = \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \Big |_{\mathbf{x}=\mathbf{x}_0},  \]  then  $  \mathrm{Law}\left(g(\mathbf{x}_0) \middle| (X, Y)\right) = \mathcal{N}(J^T \bigl(K + \tilde{\sigma}^2 I\bigr)^{-1} Y, \, B - J^T \bigl(K + \tilde{\sigma}^2I)^{-1} J),  $  where  \[  J^T = \Big [ \frac{\partial k (\mathbf{x}_0 - \mathbf{x}_1)}{\partial \mathbf{x}_0} , ... , \frac{\partial k (\mathbf{x}_0 - \mathbf{x}_n)}{\partial \mathbf{x}_0} \Big ],  \]    \[  B = \begin{bmatrix}  cov(g_1(\mathbf{x}_0),g_1(\mathbf{x}_0)) & .&.&. & cov(g_1(\mathbf{x}_0),g_m(\mathbf{x}_0)) \\  . & . & & & .\\  . & & . & & .\\  . & & & . & .\\  cov(g_m(\mathbf{x}_0),g_1(\mathbf{x}_0)) & .&.&. & cov(g_m(\mathbf{x}_0),g_m(\mathbf{x}_0)) \\  \end{bmatrix},  \]  \[  cov(g_i, g_j) = \frac{\partial^2 k (\mathbf{x}_0, \mathbf{x}_0)}{\partial x^i \partial x^j},  \]  $g_i$ --- $i$th component of gradient vector $\mathbf{g}$.