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

This is the authors' abstract. We don't add key points for this paper.

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">Lᵖ</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≤ p≤ 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">$p > 2$</tex>, it is demonstrated that there are infinitely many priors that can induce such an estimator.