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

Consider the task of estimating a random vector <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> from noisy observations <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Y=X+Z$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Z$</tex> is a standard normal vector, under the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$L^{p}$</tex> fidelity criterion. This work establishes that, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$1\leq p\leq 2$</tex>, the optimal Bayesian estimator is linear and positive definite if and only if the prior distribution on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$X$</tex> is a (non-degenerate) multivariate Gaussian. Furthermore, for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p &gt; 2$</tex>, it is demonstrated that there are infinitely many priors that can induce such an estimator.