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
for some positive number θ representing the population mutation rate, whenever a1, ..., ak is a sequence of nonnegative integers such that
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.)
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.
- 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.