Ewens's sampling formula

In population genetics, Ewens's sampling formula describes the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.

Definition
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


 * $$\operatorname{Pr}(a_1,\dots,a_n; \theta)={n! \over \theta(\theta+1)\cdots(\theta+n-1)}\prod_{j=1}^n{\theta^{a_j} \over j^{a_j} a_j!},$$

for some positive number θ representing the population mutation rate, whenever $$a_1, \ldots, a_n$$ is a sequence of nonnegative integers such that


 * $$a_1+2a_2+3a_3+\cdots+na_n=\sum_{i=1}^{n} i a_i = n.\,$$

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.

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.

Mathematical properties
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.