\Phi v\big\|_{D}/\|v\|_{K^{-1}}=\sup_{v\in \R^n,\, \Phi v=0}\|v\|_{D}/\| v\|_{K^{-1}}$. In particular, if $x$ is the solution of and $y$ the vector in , then $\big\|x-\Psi y\big\|_{D}/\|b\|_{Q^{-1}}\leq \sup_{v\in \R^n,\, \Phi v=0} \|v\|_{D}/\| v\|_{K^{-1}}$ and the right hand side is the smallest constant for which the inequality holds (for all $b$).

A simple calculation (based on the reproducing Kernel property $\<v,K_{\cdot,i}\>_{K^{-1}}=v_i$) shows that if $x$ is the solution of and $y$ the vector in , then\
$\Big|(x-\Psi y)_i\Big| \leq \big(K^\Phi_{i,i}\big)^\frac{1}{2}  \|b\|_{Q^{-1}}$, i.e. the variance of the $i$th entry of the solution of the stochastic system conditioned on $\Phi X=y$, controls the accuracy of the approximation of the $i$th entry of the solution of the deterministic system . In that sense, the role of $K^\Phi$ is analogous to that of the power function in radial basis function interpolation [@Wendland:2005; @Fasshauer:2005] and that of the Kriging function [@WuSchback:93] in geostatistics [@Myers:1992].

Energy norm estimates and selection of the prior {#subsecpriorselection}
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We will from now on assume that $A$ is symmetric positive definite. Observe that in this situation the energy norm $\|\cdot\|_A$ is of practical significance for quantifying the approximation error and Theorem \[thm:gedgdgjdh\] leads to the estimate $\|x-\Psi y\|_{K^{-1}} =\min_{z\in \R^m} \|Q^{-\frac{1}{2}}b- Q^{-\frac{1}{2}} A^{\frac{1}{2}} K^\frac{1}{2}\Phi^T z\|$ 