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.

翻译：考虑在$L^p$保真度准则下，从含噪观测$Y = X + Z$（其中$Z$为标准正态向量）中估计随机向量$X$的任务。本文证明：对于$1 \leq p \leq 2$，当且仅当$X$的先验分布为（非退化）多元高斯分布时，最优贝叶斯估计量为线性且正定。进一步，对于$p > 2$，研究表明存在无穷多个先验分布能够诱导此类估计量。