Needles and straw in a haystack: Robust confidence for possibly sparse sequences
Needles and straw in a haystack: Robust confidence for possibly sparse sequences
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大海捞针和稻草：可能稀疏序列的稳健置信度
作者：
E. Belitser;N. Nurushev
在一般的信号+噪声模型中，我们构建了一个经验贝叶斯后验，然后我们使用它来对未知的，可能是稀疏的信号进行不确定性量化。我们引入了一个新的过度偏置限制（EBR）条件，这引起了一个新的切片的整个空间，是适合不确定性量化。在EBR和一些温和的噪声条件下，我们建立了局部（预言）最优的建议的信心球。同时，我们也得到了估计和后验压缩问题的局部最优（oracle）结果。各种稀疏类的自适应极大极小结果（也适用于估计和后验收缩问题）来自我们的局部结果。
In the general signal+noise model we construct an empirical Bayes posterior which we then use for uncertainty quantification for the unknown, possibly sparse, signal. We introduce a novel 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) optimality of the proposed confidence ball. In passing, we also get the local optimal (oracle) results for estimation and posterior contraction problems. Adaptive minimax results (also for the estimation and posterior contraction problems) over various sparsity classes follow from our local results.