Confidence sequences for sampling without replacement

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 (Cn)n=1N, 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.