Efficient computation of the joint sample frequency spectra for multiple populations

A wide range of studies in population genetics have employed the sample frequency spectrum (SFS), a summary statistic which describes the distribution of mutant alleles at a polymorphic site in a sample of DNA sequences. In particular, recently there has been growing interest in analyzing the joint SFS data from multiple populations to infer parameters of complex demographic histories, including variable population sizes, population split times, migration rates, admixture proportions, and so on. Although much methodological progress has been made, existing SFS-based inference methods suffer from numerical instability and high computational complexity when multiple populations are involved and the sample size is large. In this paper, we present new analytic formulas and algorithms that enable efficient computation of the expected joint SFS for multiple populations related by a complex demographic model with arbitrary population size histories (including piecewise exponential growth). Our results are implemented in a new software package called momi (MOran Models for Inference). Through an empirical study involving tens of populations, we demonstrate our improvements to numerical stability and computational complexity.