Bayes-optimal estimation of overlap between populations of fixed size

Measuring the overlap between two populations is, in principle, straightforward. Upon fully sampling both populations, the number of shared objects—species, taxonomical units, or gene variants, depending on the context—can be directly counted. In practice, however, only a fraction of each population’s objects are likely to be sampled due to stochastic data collection or sequencing techniques. Although methods exists for quantifying population overlap under subsampled conditions, their bias is well documented and the uncertainty of their estimates cannot be quantified. Here we derive and validate a method to rigorously estimate the population overlap from incomplete samples when the total number of objects, species, or genes in each population is known, a special case of the more general β-diversity problem that is particularly relevant in the ecology and genomic epidemiology of malaria. By solving a Bayesian inference problem, this method takes into account the rates of subsampling and produces unbiased and Bayes-optimal estimates of overlap. In addition, it provides a natural framework for computing the uncertainty of its estimates, and can be used prospectively in study planning by quantifying the tradeoff between sampling effort and uncertainty.
Author summary Understanding when two populations are composed of similar species is important for ecologists, epidemiologists, and population geneticists, and in principle it is easy: just sample the two populations, compare the sets of species identified in each, and count how many appear in both populations. In practice, however, this is difficult because sampling methods typically produce only a random subset of the total population, leaving current population overlap estimates biased. Knowing only the number of shared members between two of these partial population samples, this paper shows how we can nevertheless estimate the true overlap between the full populations, when those full populations’ sizes are known. Using Bayesian statistics, we can also compute credible intervals to produce error bars. We show that using this unbiased approach has a dramatic impact on the conclusions one might draw from previously published studies in the malaria literature, which used simple but biased methods. Because the method in this paper quantifies the tradeoff between sampling effort and uncertainty, we also show how to compute the number of samples required to ensure high-confidence results, which may be useful for planning future studies or budgeting lab reagents and time.