\title{Can one learn history from the allelic spectrum?}

\author{Simon Myers$^{a}$ \\ Charles Fefferman$^{b}$ \\ Nick Patterson$^{a}$}

\begin{abstract}
It is well known that the neutral allelic frequency spectrum of a population is affected by the history of population size. A number of authors have used this fact to infer history given observed allele frequency data. We ask whether perfect information concerning the spectrum allows precise recovery of the history, and with an explicit example show that the answer is in the negative. This implies some limitations on how informative allelic spectra can be.
\end{abstract}

Keywords: Diffusions; Allelic spectrum; Kimura

\section{Introduction}

It is well known that the neutral allelic frequency spectrum of a population is affected by the population history. A number of authors have exploited this fact so as to infer population history given observed allele frequency data \cite{Przeworski_2000, Wakeley_2001, Marth_2004, Adams_Hudson_2004, Nielsen_2004, Williamson_2005, Schaffner_2005, Voight_2005, Garrigan_Hammer_2006, Chen_2007}. This raises questions about the relationship between ancient population size changes, and the frequency spectrum at mutant sites. In particular we might ask how much information can be obtained about ancestral demography of a population by typing many unlinked markers in a sample of individuals.

As more samples are gathered, in principle it is possible to gain increasingly precise information about the frequency spectrum. Therefore, here we examine two related questions regarding the history of a population. First, how much information can one obtain about population history given a sample of $n$ individuals? Second, suppose we knew the frequency spectrum exactly. Is the history of past population sizes then completely determined?

Throughout this paper, we make two simplifying assumptions. First, we assume that the population is panmictic and large enough to allow drift to be approximated by a diffusion process. Second, we assume that mutation is rare in the population so that the infinite sites model is appropriate, and we can regard every segregating site as having arisen from a unique mutation event. We consider the frequency spectrum $f$ of the derived (mutant) allele, where $f(y)$ is defined as the expected frequency of mutations at population frequency $y$ within the genome. All mutations and alleles in this paper are considered to be selectively neutral, so `frequency spectrum' would more exactly be described as a `neutral frequency spectrum'. We will regard the frequency spectrum as only defined up to an arbitrary normalizing constant. This implies that the mutation rate (assumed constant over time) is not relevant.

We ask, given the frequency spectrum $f$, is the history of past population sizes determined? This question turns out to be formally similar to the famous question \textit{Can one hear the shape of a drum?} \cite{Gordon_1992}, and like that question the answer is in the negative.

\section{Preliminary theory}

Consider a panmictic population, assumed for now to be of constant large size $N$. Under several different population genetic models, for example the discrete time Wright--Fisher model, and the continuous time Moran model, the frequency