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The sample frequency spectrum (SFS) is a widely-used summary statistic ofgenomic variation in a sample of homologous DNA sequences. It provides a highlyefficient dimensional reduction of large-scale population genomic data and itsmathematical dependence on the underlying population demography is wellunderstood, thus enabling the development of efficient inference algorithms.However, it has been recently shown that very different population demographiescan actually generate the same SFS for arbitrarily large sample sizes. Althoughin principle this nonidentifiability issue poses a thorny challenge tostatistical inference, the population size functions involved in thecounterexamples are arguably not so biologically realistic. Here, we revisitthis problem and examine the identifiability of demographic models under therestriction that the population sizes are piecewise-defined where each piecebelongs to some family of biologically-motivated functions. Under thisassumption, we prove that the expected SFS of a sample uniquely determines theunderlying demographic model, provided that the sample is sufficiently large.We obtain a general bound on the sample size sufficient for identifiability;the bound depends on the number of pieces in the demographic model and also onthe type of population size function in each piece. In the cases ofpiecewise-constant, piecewise-exponential and piecewise-generalized-exponentialmodels, which are often assumed in population genomic inferences, we provideexplicit formulas for the bounds as simple functions of the number of pieces.Lastly, we obtain analogous results for the "folded" SFS, which is often usedwhen there is ambiguity as to which allelic type is ancestral. Our results areproved using a generalization of Descartes' rule of signs for polynomials tothe Laplace transform of piecewise continuous functions.

Descartes' rule of signs and the identifiability of population demographic models from genomic variation data