Ewens's sampling formula

Ewens's sampling formula, introduced by Warren Ewens, states that under certain conditions (specified below), if a random sample of n gametes is taken from a population and classified according to the gene at a particular locus then the probability that there are a₁ alleles represented once in the sample, and a₂ alleles represented twice, and so on, is

${\displaystyle \operatorname {Pr} (a_{1},\dots ,a_{n};\theta )={n! \over \theta (\theta +1)\cdots (\theta +n-1)}\prod _{j=1}^{n}{\theta ^{a_{j}} \over j^{a_{j}}a_{j}!},}$

for some positive number θ representing the population mutation rate, whenever ${\displaystyle a_{1},\ldots ,a_{n}}$ is a sequence of nonnegative integers such that

${\displaystyle a_{1}+2a_{2}+3a_{3}+\cdots +na_{n}=\sum _{i=1}^{n}ia_{i}=n.\,}$

The phrase "under certain conditions" used above is made precise by the following assumptions:

• The sample size n is small by comparison to the size of the whole population; and
• The population is in statistical equilibrium under mutation and genetic drift and the role of selection at the locus in question is negligible; and
• Every mutant allele is novel.
  See also: Infinite-alleles model

This is a probability distribution on the set of all partitions of the integer n. Among probabilists and statisticians it is often called the multivariate Ewens distribution.

When θ = 0, the probability is 1 that all n genes are the same. When θ = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed random permutation. As θ → ∞, the probability that no two of the n genes are the same approaches 1.

This family of probability distributions enjoys the property that if after the sample of n is taken, m of the n gametes are chosen without replacement, then the resulting probability distribution on the set of all partitions of the smaller integer m is just what the formula above would give if m were put in place of n.

The Ewens distribution arises naturally from the Chinese restaurant process.

• Chinese restaurant table distribution
• Coalescent theory
• Unified neutral theory of biodiversity
• Biomathematics

• Warren Ewens, "The sampling theory of selectively neutral alleles", Theoretical Population Biology, volume 3, pages 87–112, 1972.
• H. Crane. (2016) "The Ubiquitous Ewens Sampling Formula", Statistical Science, 31:1 (Feb 2016). This article introduces a series of seven articles about Ewens Sampling in a special issue of the journal.
• J.F.C. Kingman, "Random partitions in population genetics", Proceedings of the Royal Society of London, Series B, Mathematical and Physical Sciences, volume 361, number 1704, 1978.
• S. Tavare and W. J. Ewens, "The Multivariate Ewens distribution." (1997, Chapter 41 from the reference below).
• N.L. Johnson, S. Kotz, and N. Balakrishnan (1997) Discrete Multivariate Distributions, Wiley. ISBN 0-471-12844-9.