Movatterモバイル変換

[0]ホーム

Jump to content

Ewens's sampling formula

Català

Edit links

From Wikipedia, the free encyclopedia

Sampling formula which describes the probabilities of alleles in a sample

This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(August 2011) (Learn how and when to remove this message)

Inpopulation genetics,Ewens's sampling formula describes theprobabilities associated with counts of how many differentalleles are observed a given number of times in thesample.

Definition

[edit]

Ewens's sampling formula, introduced byWarren Ewens, states that under certain conditions (specified below), if a random sample ofngametes is taken from a population and classified according to thegene at a particularlocus then theprobability that there area₁alleles represented once in the sample, anda₂ 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 numberθ representing thepopulation 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}ia_{i}=n.\,

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

The sample sizen is small by comparison to the size of the whole population; and
The population is in statistical equilibrium undermutation andgenetic 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 aprobability distribution on the set of allpartitions of the integern. Among probabilists and statisticians it is often called themultivariate Ewens distribution.

Mathematical properties

[edit]

Whenθ = 0, the probability is 1 that alln genes are the same. Whenθ = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributedrandom permutation. Asθ → ∞, the probability that no two of then genes are the same approaches 1.

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

The Ewens distribution arises naturally from theChinese restaurant process.

Notes

[edit]

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.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling'sT-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher'sz Kaniadakis κ-Gaussian Gaussianq Generalized hyperbolic Generalized logistic (logistic-beta) Generalized normal Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson'sS_U Landau Laplace Asymmetric Logistic Noncentralt Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student'st Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakisκ-exponential Kaniadakisκ-Gamma Kaniadakisκ-Weibull Kaniadakisκ-Logistic Kaniadakisκ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda