European Institute for Statistics, Probability, Stochastic Operations Research
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Needles and Straw in a Haystack: Robust Empirical Bayes Confidence for Possibly Sparse Sequences
Nurzhan Nurushev, VU Amsterdam

In the signal+noise model (the noise is not necessarily independent normals) we construct an empirical Bayes posterior which we then use for \emph{uncertainty quantification} for the unknown, possibly sparse, signal. We introduce a novel \emph{excessive bias restriction} (EBR) condition, which gives rise to a new slicing of the entire space that is suitable for uncertainty quantification. Under EBR and some mild conditions on the noise, we establish the local (oracle) confidence optimality of the empirical Bayes credible ball. In passing, we also get the local optimal results for estimation and posterior contraction problems. Adaptive minimax results (also for the estimation and posterior contraction problems) over sparsity classes follow from our local results.

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Eurandom 2012