Fundamental limits on the accuracy of demographic inference based on the sample frequency spectrum
Significance Numerous empirical studies in population genetics have used a summary statistic called the sample frequency spectrum (SFS) which summarizes the information in a sample of DNA sequences. Despite their popularity, the accuracy of inference methods based on the SFS is difficult to characterize theoretically, and it is currently unknown how the estimation accuracy improves as more sites in the genome are used. Here, we establish information theoretic limits on the accuracy of all estimators that use the SFS to infer population size histories. We study the rate of convergence to the true answer as the amount of data increases, and obtain the surprising result that it is exponentially worse than known convergence rates for many classical estimation problems in statistics. The sample frequency spectrum (SFS) of DNA sequences from a collection of individuals is a summary statistic that is commonly used for parametric inference in population genetics. Despite the popularity of SFS based inference methods, little is currently known about the information theoretic limit on the estimation accuracy as a function of sample size. Here, we show that using the SFS to estimate the size history of a population has a minimax error of at least O(1/log s) where s is the number of independent segregating sites used in the analysis. This rate is exponentially worse than known convergence rates for many classical estimation problems in statistics. Another surprising aspect of our theoretical bound is that it does not depend on the dimension of the SFS, which is related to the number of sampled individuals. This means that, for a fixed number s of segregating sites considered, using more individuals does not help to reduce the minimax error bound. Our result pertains to populations that have experienced a bottleneck, and we argue that it can be expected to apply to many populations in nature.
Proceedings of the National Academy of Sciences