Papers › Confidence sequences for sampling without replacement
Confidence sequences for sampling without replacement
Ian Waudby-Smith, Aaditya Ramdas
Many practical tasks involve sampling sequentially without replacement (WoR) from a finite population of size N, in an attempt to estimate some parameter θ^⋆. 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^⋆. A CS is a sequence of confidence sets (Cₙ)ₙ₌₁ᴺ, that shrink in size, and all contain θ^⋆ 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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