given $\Phi X=y$ is then obtained by observing that for $v\in \R^n$, $v^T K^{\Phi} v=\E\big[|v^T X-  v^T \Psi \Phi X|^2\big]= v^T K v -  v^T \Psi \Phi K v$.

Variational/optimal recovery properties and approximation error {#subsecvarprop}
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For a $n\times n$ symmetric positive definite matrix $M$ let $\<\cdot,\cdot\>_{M}$ be the (scalar) product on $\R^n$ defined by: for $u, v\in \R^n$, $ \<u, v\>_{M} := u^T M v$ and write $\|v\|_{M}:=\<v, v\>_M^\frac{1}{2}$ the corresponding norm. When $M$ is the identity matrix then we write $\<u, v\>$ and $ \|v\|$ the corresponding scalar product and norm. For a linear subspace $V$ of $\R^n$ we write $P_{V, M}$ for the orthogonal projections onto $V$ with respect to the scalar product $\<\cdot,\cdot\>_{M}$. For a (possibly rectangular) matrix $B$ we write $\Img(B)$ the image (range) of $B$ and $\Ker(B)$ the null space of $B$. For an integer $n$, let $I_n$ be the $n\times n$ identity matrix.

\[lem:minimizingproperty\] For $w\in \R^m$, $\Psi w$ is the unique minimizer of the following quadratic problem $$\label{eq:dueihdbis}
\begin{cases}
\text{Minimize }  &\<v, v\>_{K^{-1}}\\
\text{Subject to } &\Phi v=w\text{ and } v \in \R^n\,.
\end{cases}$$ In particular, $v=\Psi y$, the Bayesian approximation of the solution of , is the unique minimizer of 