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 $1\le p\le 2$, 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.