Title: Estimating the unseen from multiple populations,
Abstract: Given samples from a distribution, how many new elements should we expect to
find if we continue sampling this distribution? This is an important and
actively studied problem, with many applications ranging from unseen species
estimation to genomics. We generalize this extrapolation and related unseen
estimation problems to the multiple population setting, where population $j$
has an unknown distribution $D_j$ from which we observe $n_j$ samples. We
derive an optimal estimator for the total number of elements we expect to find
among new samples across the populations. Surprisingly, we prove that our
estimator's accuracy is independent of the number of populations. We also
develop an efficient optimization algorithm to solve the more general problem
of estimating multi-population frequency distributions. We validate our methods
and theory through extensive experiments. Finally, on a real dataset of human
genomes across multiple ancestries, we demonstrate how our approach for unseen
estimation can enable cohort designs that can discover interesting mutations
with greater efficiency.