L1 Estimation: On the Optimality of Linear Estimators

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.