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

:$\backslash operatorname\{Pr\}(a\_1,\backslash dots,a\_n;\; \backslash theta)=\{n!\; \backslash over\; \backslash theta(\backslash theta+1)\backslash cdots(\backslash theta+n-1)\}\backslash prod\_\{j=1\}^n\{\backslash theta^\{a\_j\}\; \backslash over\; j^\{a\_j\}\; a\_j!\},$

for some positive number θ representing the population mutation rate, whenever $a\_1,\; \backslash ldots,\; a\_n$ is a sequence of nonnegative integers such that

:$a\_1+2a\_2+3a\_3+\backslash cdots+na\_n=\backslash sum\_\{i=1\}^\{n\}\; i\; a\_i\; =\; n.\backslash ,$

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) "[https://www.researchgate.net/publication/280311472_The_Ubiquitous_Ewens_Sampling_Formula 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}}.

{{ProbDistributions|multivariate}}

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Category:Theory of probability distributions

Category:Population genetics

Category:Discrete distributions