arXiv 2006.043472020-06stat.MEmath.STstat.MLstat.TH

无放回不放回抽样的置信序列

Confidence sequences for sampling without replacement

Ian Waudby-Smith, Aaditya Ramdas

摘要：

许多实际任务涉及从大小为$N$的有限总体中顺序无替换（WoR）抽样，以尝试估计某些参数$θ^\star$。在整个过程中准确量化不确定性是一项重要的任务，但也是必要的，因为它通常决定我们何时停止收集样本并自信地报告结果。我们提出了一套工具，设计置信序列（CS）的$θ^\star$。CS是置信集$(C_n)_{n=1}^N$的序列，其大小缩小，并且所有置信集都以高概率同时包含$θ^\star$。我们提出了一个通用的方法来构建一个频率论CS使用贝叶斯工具，基于这样一个事实，即在地面真理的比例之前的后验是一个鞅。然后，我们提出了Hoeffding-和mixical-Bernstein型时间均匀CS和固定时间置信区间的采样WoR，提高了以前的界限在文献中，明确量化的好处WoR采样。

英文摘要：

Many practical tasks involve sampling sequentially without replacement (WoR) from a finite population of size $N$, in an attempt to estimate some parameter $θ^\star$. 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 $θ^\star$. A CS is a sequence of confidence sets $(C_n)_{n=1}^N$, that shrink in size, and all contain $θ^\star$ simultaneously with high probability. We present a generic approach to constructing a frequentist CS using Bayesian tools, based on the fact that the ratio of a prior to the posterior at the ground truth is a martingale. 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 and explicitly quantify the benefit of WoR sampling.

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英文摘要：

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