  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.
