In large-scale studies with parallel signal-plus-noise observations, the local false discovery rate is a summary statistic that is often presumed to be equal to the posterior probability that the signal is null. We prefer to call the latter quantity the local null-signal rate to emphasize our view that a null signal and a false discovery are not identical events. The local null-signal rate is commonly estimated through empirical Bayes procedures that build on the `zero density assumption', which attributes the density of observations near zero entirely to null signals. In this paper, we argue that this strategy does not furnish estimates of the local null-signal rate, but instead of a quantity we call the complementary local activity rate (clar). Although it is likely to be small, an inactive signal is not necessarily zero. The local activity rate addresses two shortcomings of the local null-signal rate. First, it is a weakly continuous functional of the signal distribution, and second, it takes on sensible values when the signal is sparse but not exactly zero. Our findings clarify the interpretation of local false-discovery rates estimated under the zero density assumption.

翻译：在大规模并行信号加噪声观测研究中，局部错误发现率通常被假定为信号为零的后验概率。我们倾向于将后者称为局部零信号率，以强调零信号与错误发现并非同一事件。局部零信号率通常通过基于"零密度假设"的经验贝叶斯方法进行估计——该假设将观测值接近零的密度完全归因于零信号。本文论证这种策略提供的并非局部零信号率的估计，而是我们称之为互补局部活动率（clar）的统计量。尽管非活跃信号可能很小，但并不必然为零。局部活动率克服了局部零信号率的两个缺陷：其一，它是信号分布的弱连续泛函；其二，当信号稀疏但非严格为零时，该统计量能给出合理取值。我们的发现阐明了在零密度假设下估计的局部错误发现率的解释含义。