beta distribution

English

Probability density function of the beta distribution for various values of α and β

Noun

beta distribution (plural beta distributions)

1. (probability, statistics) Any of a family of continuous probability distributions defined on the interval [0, 1] whose shape is parametrised by two positive parameters, denoted α and β, which appear as exponents in the associated random variable.
   • 2006, Kent E. Holsinger, “Chapter 2: Bayesian hierarchical models in geographical genetics”, in James S. Clark, Alan E. Gelfand, editors, Hierarchical Modelling for the Environmental Sciences: Statistical Methods and Applications, Oxford University Press, page 25:

     In particular, I illustrate how a hierarchical model in which the allele frequency distribution among populations is approximated by a Beta distribution and distribution of mean allele frequencies across loci is also approximated by a Beta distribution, accounts for the time-correlation of allele frequencies that existing methods of inference ignore.

   • 2014, Stephen F. Bush, Smart Grid: Communication-Enabled Intelligence for the Electric Power Grid, Wiley, 2015, Reprint with corrections, page 366,

     When more than two mutually exclusive events are involved in a frame of reference, the beta distributions form the marginals of a Dirichlet distribution.

   • 2014, Nikolay Tcholtchev, Ina Schieferdecker, Framework for Ensuring Runtime Stability of Control Loops in Multi-agent Networked Environments, Marina Gavrilova, C. J. Kenneth Tan (editors), Transactions on Computational Science XXII, Springer, LNCS 8360, page 80,

     We developed a Matlab script that implements the machine learning procedures as specified by (11), as well as by (13) with applying the beta-distribution (14) as a probability measure.

Usage notes

Of importance in Bayesian inference as the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions. (That is, it can be used, among other things, to describe initial knowledge of probability of success.)

Mathematically, the probability density function, for 0 ≤ x ≤ 1, can be expressed as $\textstyle f(x;\alpha ,\beta )={\frac {1}{\mathrm {B} (\alpha ,\beta )}}x^{\alpha -1}(1-x)^{\beta -1}$, where the beta function B is a normalisation constant that ensures the probability function integrates to 1 over the interval.

Synonyms

• (any of a family of distributions): beta distribution of the first kind

Derived terms

• beta distribution of the second kind
• beta prime distribution

Translations

type of probability distribution

• Finnish: beta-jakauma, β-jakauma

Further reading

• Bayesian inference on Wikipedia.Wikipedia
• Conjugate prior on Wikipedia.Wikipedia
• Dirichlet distribution on Wikipedia.Wikipedia