We present novel bounds for estimating discrete probability distributions under the $\ell_\infty$ norm. These are nearly optimal in various precise senses, including a kind of instance-optimality. Our data-dependent convergence guarantees for the maximum likelihood estimator significantly improve upon the currently known results. A variety of techniques are utilized and innovated upon, including Chernoff-type inequalities and empirical Bernstein bounds. We illustrate our results in synthetic and real-world experiments. Finally, we apply our proposed framework to a basic selective inference problem, where we estimate the most frequent probabilities in a sample.

翻译：我们提出了在$\ell_\infty$范数下估计离散概率分布的新界。这些界限在多种精确意义上近乎最优，包括一种实例最优性。我们针对最大似然估计器提出的数据依赖收敛保证显著改进了现有已知结果。本文运用并创新了多种技术，包括Chernoff型不等式和经验Bernstein界。我们通过合成实验和真实世界实验展示了结果。最后，我们将所提出的框架应用于一个基础的选择性推断问题，在样本中估计出现频率最高的概率。