Consider the problem of estimating a random variable $X$ from noisy observations $Y = X+ Z$, where $Z$ is standard normal, under the $L^1$ fidelity criterion. It is well known that the optimal Bayesian estimator in this setting is the conditional median. This work shows that the only prior distribution on $X$ that induces linearity in the conditional median is Gaussian. Along the way, several other results are presented. In particular, it is demonstrated that if the conditional distribution $P_{X|Y=y}$ is symmetric for all $y$, then $X$ must follow a Gaussian distribution. Additionally, we consider other $L^p$ losses and observe the following phenomenon: for $p \in [1,2]$, Gaussian is the only prior distribution that induces a linear optimal Bayesian estimator, and for $p \in (2,\infty)$, infinitely many prior distributions on $X$ can induce linearity. Finally, extensions are provided to encompass noise models leading to conditional distributions from certain exponential families.

翻译：考虑在$L^1$保真度准则下，从含噪观测$Y = X + Z$（其中$Z$为标准正态分布）估计随机变量$X$的问题。众所周知，此场景下的最优贝叶斯估计量为条件中位数。本文证明，使条件中位数呈现线性的唯一先验分布是高斯分布。在此过程中，还给出了若干其他结果。特别地，研究表明：若对所有$y$，条件分布$P_{X|Y=y}$均为对称分布，则$X$必服从高斯分布。此外，我们考虑其他$L^p$损失，观察到如下现象：当$p \in [1,2]$时，高斯分布是唯一能使最优贝叶斯估计量呈线性的先验分布；而当$p \in (2,\infty)$时，存在无穷多个$X$的先验分布可产生线性性。最后，本文将结论拓展至包含特定指数族条件分布所对应的噪声模型。