which simplifies to the energy norm estimate expressed by Corollary under the choice $Q=A$ (note that $K^{-1}=A$ under that choice).

\[cor:geaaedgjdh\] If $A$ is symmetric positive definite and $Q=A$, then for $v\in \R^n$, $\|v-\Psi \Phi v\|_{A}=\min_{z\in \R^m} \|v-A^{-1}\Phi^T z\|_{A}$. Therefore, if $x$ is the solution of and $y$ the vector in , then $\|x-\Psi y\|_{A} =\min_{w\in \R^m} \|x-\Psi w\|_{A} =\min_{z\in \R^m} \|x-A^{-1}\Phi^T z\|_{A}$. In particular $$\label{eq:gdugeueydd}
\|x-\Psi y\|_{A} =\min_{z\in \R^m} \|A^{-\frac{1}{2}} b-A^{-\frac{1}{2}}\Phi^T z\|\,.$$

\[rmk:galerkinproj\] Therefore, according to Corollary \[cor:geaaedgjdh\], if $Q=A$, then $\Psi y$ is the Galerkin approximation of $x$, i.e. the best approximation of $x$ in $\|\cdot\|_A$-norm in the image of $\Psi$ (which is equal to the image of $A^{-1}\Phi^T$). This is interesting because $\Psi y$ is obtained without the prior knowledge of $b$.

Corollary \[cor:geaaedgjdh\] and Remark \[rmk:galerkinproj\] motivate us to select $Q=A$ as the covariance matrix of the Gaussian prior distribution (mixed strategy of Player II).

Impact and selection of the measurement matrix $\Phi$
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It is natural to wonder how good this recovery strategy is (under the choice $Q=A$) compared to the best possible function of $y$ and how the approximation error is impacted by the measurement matrix $\Phi$. If the energy norm is used to 