L ¹ Estimation: On the Optimality of Linear Estimators

Consider the problem of estimating a random variable X from noisy observations <tex-math notation="LaTeX">$Y = X+ Z$ </tex-math>, where Z is standard normal, under the <tex-math notation="LaTeX">L¹ </tex-math> 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 <tex-math notation="LaTeX">PX|Y=y </tex-math> is symmetric for all y, then X must follow a Gaussian distribution. Additionally, we consider other <tex-math notation="LaTeX">Lᵖ </tex-math> losses and observe the following phenomenon: for <tex-math notation="LaTeX">p ∈ [1,2] </tex-math>, Gaussian is the only prior distribution that induces a linear optimal Bayesian estimator, and for <tex-math notation="LaTeX">p ∈ (2,∞) </tex-math>, 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.