  Consider the task of estimating a random vector $X$ from noisy observations
$Y = X + Z$, where $Z$ is a standard normal vector, under the $L^p$ fidelity
criterion. This work establishes that, for $1 \leq p \leq 2$, the optimal
Bayesian estimator is linear and positive definite if and only if the prior
distribution on $X$ is a (non-degenerate) multivariate Gaussian. Furthermore,
for $p > 2$, it is demonstrated that there are infinitely many priors that can
induce such an estimator.
