Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise

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.