Multivariate Priors and the Linearity of Optimal Bayesian Estimators under Gaussian Noise
CoRR(2024)
摘要
Consider the task of estimating a random vector X from noisy observations
Y = X + Z, where Z is a standard normal vector, under the L^p fidelity
criterion. This work establishes that, for 1 ≤ p ≤ 2, the optimal
Bayesian estimator is linear and positive definite if and only if the prior
distribution on X is a (non-degenerate) multivariate Gaussian. Furthermore,
for p > 2, it is demonstrated that there are infinitely many priors that can
induce such an estimator.
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