  When is optimal estimation linear? It is well known that, when a Gaussian
source is contaminated with Gaussian noise, a linear estimator minimizes the
mean square estimation error. This paper analyzes, more generally, the
conditions for linearity of optimal estimators. Given a noise (or source)
distribution, and a specified signal to noise ratio (SNR), we derive conditions
for existence and uniqueness of a source (or noise) distribution for which the
$L_p$ optimal estimator is linear. We then show that, if the noise and source
variances are equal, then the matching source must be distributed identically
to the noise. Moreover, we prove that the Gaussian source-channel pair is
unique in the sense that it is the only source-channel pair for which the mean
square error (MSE) optimal estimator is linear at more than one SNR values.
Further, we show the asymptotic linearity of MSE optimal estimators for low SNR
if the channel is Gaussian regardless of the source and, vice versa, for high
SNR if the source is Gaussian regardless of the channel. The extension to the
vector case is also considered where besides the conditions inherited from the
scalar case, additional constraints must be satisfied to ensure linearity of
the optimal estimator.
