This paper investigates the multiple testing problem for high-dimensional sparse binary sequences motivated by the crowdsourcing problem in machine learning. We adopt an empirical Bayes approach to estimate possibly sparse sequences with Bernoulli noises. We found a surprising result that the hard thresholding rule deduced from the spike-and-slab posterior is not optimal, even using a uniform prior. Two approaches are then proposed to calibrate the posterior for achieving the optimal signal detection boundary, and two multiple testing procedures are constructed based on these calibrated posteriors. Sharp frequentist theoretical results for these procedures are obtained, showing both can effectively control the false discovery rate uniformly for signals under a sparsity assumption. Numerical experiments are conducted to validate our theory in finite samples.

翻译：本文研究了一种由机器学习中的众包问题所驱动的高维稀疏二元序列的多重检验问题。我们采用经验贝叶斯方法来估计可能带有伯努利噪声的稀疏序列。发现了一个令人惊讶的结果：即使使用均匀先验，从spike-and-slab后验推导出的硬阈值规则也并非最优。随后，我们提出了两种方法来校准后验以实现最优信号检测边界，并基于这些校准后验构建了两种多重检验程序。获得了这些程序的尖锐频率学派理论结果，表明两者在稀疏性假设下均能有效一致地控制信号的错误发现率。数值实验在有限样本下验证了我们的理论。