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The sample frequency spectrum (SFS), which describes the distribution ofmutant alleles in a sample of DNA sequences, is a widely used summary statisticin population genetics. The expected SFS has a strong dependence on thehistorical population demography and this property is exploited by popularstatistical methods to infer complex demographic histories from DNA sequencedata. Most, if not all, of these inference methods exhibit pathologicalbehavior, however. Specifically, they often display runaway behavior inoptimization, where the inferred population sizes and epoch durations candegenerate to 0 or diverge to infinity, and show undesirable sensitivity of theinferred demography to perturbations in the data. The goal of this paper is toprovide theoretical insights into why such problems arise. To this end, wecharacterize the geometry of the expected SFS for piecewise-constantdemographic histories and use our results to show that the aforementionedpathological behavior of popular inference methods is intrinsic to the geometryof the expected SFS. We provide explicit descriptions and visualizations for atoy model with sample size 4, and generalize our intuition to arbitrary samplesizes n using tools from convex and algebraic geometry. We also develop auniversal characterization result which shows that the expected SFS of a sampleof size n under an arbitrary population history can be recapitulated by apiecewise-constant demography with only k(n) epochs, where k(n) is between n/2and 2n-1. The set of expected SFS for piecewise-constant demographies withfewer than k(n) epochs is open and non-convex, which causes the above phenomenafor inference from data.

Geometry of the sample frequency spectrum and the perils of demographic inference