Empirical Bayes denoising works by estimating a prior distribution from the data, then using that prior to shrink noisy observations toward their likely true values. The method is powerful but fragile: the estimated prior is only as good as the model assumptions. If the noise is not Gaussian or the prior class is misspecified, the shrinkage can make things worse rather than better.
Koenker and Gu pursue two classical strategies for robustification. First, Hodges and Lehmann's restricted Bayes approach: instead of optimizing over all possible priors, restrict the class to those satisfying certain constraints, reducing sensitivity to any single prior specification. The answer depends less on which prior you choose because fewer priors are eligible. Second, replacing the Gaussian noise assumption with robust alternatives following Huber and Mallows — distributions with heavier tails that accommodate occasional outliers without distorting the shrinkage applied to typical observations.
Both strategies trade efficiency for stability. Under the correct model, the robust estimator is slightly worse. Under misspecification, the robust estimator degrades gracefully while the standard one can catastrophically overshrink or undershrink. The classical proposals from Huber (1964) and Mallows (1978) provide the mathematical machinery: least favorable distributions that minimax the maximum regret across a neighborhood of models.
The through-claim is about the cost of knowing the answer. Standard empirical Bayes commits to a model — Gaussian noise, unrestricted prior — and extracts the maximum information that model contains. Robust empirical Bayes deliberately discards information by restricting what it's willing to assume. The discarded information is exactly the information that doesn't survive misspecification. The robust estimator is less efficient because it refuses to learn what might be wrong.