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In population genetics, 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 Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ...
Warren Ewens is a professor of biology at the University of Pennsylvania. ...
Gametes, from the ancient Greek γαμετης (spouse), are the specialized germ cells that come together during fertilization (conception) in organisms that reproduce sexually. ...
This stylistic schematic diagram shows a gene in relation to the double helix structure of DNA and to a chromosome (right). ...
In biology and evolutionary computation, a locus is the position of a gene (or other significant sequence) on a chromosome. ...
The word probability derives from the Latin probare (to prove, or to test). ...
An allele is any one of a number of viable DNA codings of the same gene (sometimes the term refers to a non-gene sequence) occupying a given locus (position) on a chromosome. ...
 for some positive number θ, whenever a1, ..., an is a sequence of nonnegative integers such that  The phrase "under certain conditions", used above, must of course be made precise. The assumptions are (1) the sample size n is small by comparison to the size of the whole population, and (2) the population is in statistical equilibrium under mutation and genetic drift and the role of selection at the locus in question is negligible, and (3) every mutant allele is novel. (See also idealised population.) In biology, mutations are changes to the genetic material (usually DNA or RNA). ...
Genetic drift is the term used in population genetics to refer to the statistical drift over time of allele frequencies in a finite population due to random sampling effects in the formation of successive generations. ...
In population genetics an idealised population or a Fisher — Wright population is a population whose members can mate and reproduce with any other member of the other gender, and where random genetic drift does not occur. ...
This is a probability distribution on the set of all partitions of the integer n. Among probabilists and statisticians it is often called the Ewens distribution. In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...
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  the probability that no two of the n genes are the same approaches 1. A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. ...
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.
References
- Warren Ewens, "The sampling theory of selectively neutral alleles", Theoretical Population Biology, volume 3, pages 87—112, 1972.
- J.F.C. Kingman, "Random partitions in population genetics", Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, volume 361, number 1704, 1978.
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