Fundamental limits on the accuracy of demographic inference based on the sample frequency spectrum

TL;DR

This paper establishes fundamental information-theoretic limits on the accuracy of demographic history inference from the sample frequency spectrum, showing that increasing sample size does not necessarily improve estimation accuracy.

Contribution

It provides the first minimax error bounds for SFS-based demographic inference, revealing intrinsic limitations independent of sample size.

Findings

Minimax error for population size history estimation is at least O(1/ log s).

Increasing the number of sampled individuals does not reduce the error bound.

The results apply to populations with bottlenecks and are likely relevant to many natural populations.

Abstract

The sample frequency spectrum (SFS) of DNA sequences from a collection of individuals is a summary statistic which is commonly used for parametric inference in population genetics. Despite the popularity of SFS-based inference methods, currently little is 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.