Confidence sequences for sampling without replacement
Confidence sequences for sampling without replacement
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发表时间：
2020-06
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通讯作者：
Ian Waudby-Smith;Aaditya Ramdas
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作者：
Ian Waudby-Smith;Aaditya Ramdas
许多实际任务涉及从大小为\(N\)的有限总体中无放回地依次抽样（WoR），以尝试估计某个参数\(\theta^\star\)。在这个过程中准确地量化不确定性是一项不平凡的任务，但却是必要的，因为它常常决定我们何时停止收集样本并自信地报告结果。我们提出了一套用于为\(\theta^\star\)设计置信序列（CS）的工具。置信序列是一列置信集\((C_n)_{n = 1}^N\)，其大小会收缩，并且所有置信集都以高概率同时包含\(\theta^\star\)。我们首先利用贝叶斯后验与鞅之间的关系，为超几何分布的参数构建一个（频率主义的）置信序列。然后，我们给出了针对无放回抽样的霍夫丁型和经验伯恩斯坦型时间一致置信序列以及固定时间置信区间，它们改进了文献中先前的界。
Many practical tasks involve sampling sequentially without replacement (WoR) from a finite population of size $N$, in an attempt to estimate some parameter $\theta^\star$. Accurately quantifying uncertainty throughout this process is a nontrivial task, but is necessary because it often determines when we stop collecting samples and confidently report a result. We present a suite of tools for designing confidence sequences (CS) for $\theta^\star$. A CS is a sequence of confidence sets $(C_n)_{n=1}^N$, that shrink in size, and all contain $\theta^\star$ simultaneously with high probability. We first exploit a relationship between Bayesian posteriors and martingales to construct a (frequentist) CS for the parameters of a hypergeometric distribution. We then present Hoeffding- and empirical-Bernstein-type time-uniform CSs and fixed-time confidence intervals for sampling WoR which improve on previous bounds in the literature.