This paper develops a framework for differentially private $e$-values under Gaussian differential privacy ($μ$-GDP). We characterize the canonical noise mechanism, establishing that optimal multiplicative perturbation follows a Gaussian distribution. Using this distribution, we derive a globally sharp rejection threshold that strictly improves upon the standard Markov bound. Asymptotic analysis shows that in low-sensitivity regimes, the calibrated private test achieves a net power gain over the non-private baseline. For multiple testing, we introduce a recursive peeling algorithm that adaptively concentrates the privacy budget on the most promising hypotheses. This construction guarantees rigorous $μ$-GDP and yields valid private $e$-values compatible with standard multiple testing procedures. Simulations and a genome-wide association study confirm that the method controls the false discovery rate while improving upon naive all-noisy privatization and recovering power close to non-private benchmarks.

翻译：本文在$μ$-高斯差分隐私框架下，发展了差分隐私$e$-值的理论体系。我们刻画了典范噪声机制，证明最优乘性扰动服从高斯分布。基于该分布推导出全局尖锐拒绝阈值，该阈值严格优于标准马尔可夫界。渐近分析表明，在低敏感度情况下，校准后的私有检验相较于非私有基线可获得净功效增益。针对多重检验问题，我们提出递归剥离算法，该算法能自适应地将隐私预算集中于最有希望的假设上。该构造保证了严格的$μ$-GDP性质，并生成与标准多重检验程序兼容的有效私有$e$-值。模拟实验与全基因组关联研究证实，该方法在控制错误发现率的同时，相较于朴素全噪声私有化方案具有更优表现，其检验功效接近非私有基准。