L^1 Estimation: On the Optimality of Linear Estimators
arXiv (Cornell University)(2023)
摘要
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 ∈
[1,2], Gaussian is the only prior distribution that induces a linear optimal
Bayesian estimator, and for p ∈ (2,∞), 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.
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关键词
linear estimators,estimation,optimality
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