x =y,$$ where $\Phi$ (the measurement matrix) is a known, rank $m$, $m\times n$ real matrix such that $m<n$ and $y$ (the measurement vector) is a known vector of $\R^m$, and (2) the norm $b^T T^{-1} b$ of $b$ is known or bounded by a known constant (e.g., $b^T T^{-1} b \leq 1$), where $T^{-1}$ is a known positive definite $n\times n$ matrix (with $T^{-1}$ being the identity matrix as a prototypical example). Observe that since $m<n$, the measurements are, a priori, not sufficient to recover the exact value $x$.

As described in Section \[secdt\], by formulating this recovery problem as a (non-cooperative) information game (where Player I chooses $b$ and Player II chooses an approximation $x^*$ of $x$ based on the observation $\Phi x$), one (Player II) is naturally lead to search for mixed strategy in the Bayesian class by placing a prior distribution on $b$. The purpose of this section is to analyze the resulting approximation error and select the prior distribution accordingly. To preserve the linearity (i.e. simplicity and computational efficiency) of calculations we will restrict Player II’s decision space to Gaussian priors.

Player I’s mixed strategy
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We will therefore, in the first step of the 