analysis, replace $b$ in by $\xi$, a centered Gaussian vector of $\R^n$ with covariance matrix $Q$ (which may be distinct from $T$) and consider the following stochastic linear system $$\label{eqn:noisy}
A X=\xi\,.$$ The Bayesian answer (a mixed strategy for Player II) to the recovery problem of Section \[sec:problem\] is to approximate $x$ by the conditional expectation $\E[X|\Phi X=y]$.

\[thm:Gaussian\] The solution $X$ of is a centered Gaussian vector of $\R^n$ with covariance matrix $$\label{eqhbddjdhh}
K=A^{-1} Q (A^{-1})^T\,.$$ Furthermore, $X$ conditioned on the value $\Phi X=y$ is a Gaussian vector of $\R^n$ with mean $\E[X|\Phi X=y]= \Psi y$, and of covariance matrix $K^{\Phi}$, where $\Psi$ is the $n \times m$ matrix $$\label{eq:phiidef}
\Psi:=K \Phi^T (\Phi K \Phi^T)^{-1},$$ and $K^{\Phi}$ is the rank $n-m$ positive $n\times n$ symmetric matrix defined by $K^{\Phi}:=  K  -   \Psi \Phi K$.

simply follows from $X=A^{-1}\xi$. Since $X$ is a Gaussian vector, $\E[X|\Phi X=y]= \Psi y$ where $\Psi$ is a $n\times m$ matrix minimizing the mean squared error $\E\big[|X- M\Phi X|^2\big]$ over all $n\times m$ matrices $M$. We have $\E\big[|X- M\Phi X|^2\big]= \Tr[K]+ \Tr[M \Phi K \Phi^T M^T] -2 \Tr[\Phi K M]$ whose minimum is achieved for $M=\Psi$ as defined by . The covariance matrix of $X$ 