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 $Y = X + Z$ 中估计随机变量 $X$ 的问题，其中 $Z$ 服从标准正态分布。众所周知，该设定下的最优贝叶斯估计量是条件中位数。本文证明，使条件中位数具有线性性的唯一先验分布是高斯分布。在此过程中，我们还给出了若干其他结果。特别地，若条件分布 $P_{X|Y=y}$ 对所有 $y$ 均对称，则 $X$ 必服从高斯分布。此外，我们考察了其他 $L^p$ 损失函数，并观察到以下现象：当 $p \in [1,2]$ 时，高斯分布是唯一使最优贝叶斯估计量具有线性性的先验分布；而当 $p \in (2,\infty)$ 时，存在无穷多个 $X$ 的先验分布可诱导线性性。最后，我们将结果推广至包含来自特定指数族条件分布的噪声模型。