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 ≤ p ≤ 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.