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$Z$服从标准正态分布）中估计随机变量$X$的问题。众所周知，在此设置下的最优贝叶斯估计器是条件中位数。本研究表明，能诱导条件中位数呈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，证明了若对所有$y$，条件分布$P_{X|Y=y}$均对称，则$X$必服从高斯分布。此外，我们考虑了其他$L^p$损失，并观察到以下现象：对于$p \in [1,2]$，高斯分布是唯一能诱导最优贝叶斯估计器呈线性的先验分布；而对于$p \in (2,\infty)$，存在无穷多种$X$的先验分布可诱导线性性。最后，研究结果被推广至涵盖能产生来自特定指数族条件分布的噪声模型。