Beta-binomial distribution

In probability theory and statistics, the beta-binomial distribution is a family of discrete probability distributions on a finite support arising when the probability of success in each of a fixed or known number of Bernoulli trials is either unknown or random. It is frequently used in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed binomial distribution.

It reduces to the Bernoulli distribution as a special case when n = 1. For α = β = 1, it is the discrete uniform distribution from 0 to n. It also approximates the binomial distribution arbitrarily well for large α and β. The beta-binomial is a one-dimensional version of the multivariate Pólya distribution, as the binomial and beta distributions are special cases of the multinomial and Dirichlet distributions, respectively.

Motivation and derivation

Beta-binomial distribution as a compound distribution

The Beta distribution is a conjugate distribution of the binomial distribution. This fact leads to an analytically tractable compound distribution where one can think of the <math> p </math> parameter in the binomial distribution as being randomly drawn from a beta distribution. Namely, if

<math>

begin L(k|p) & = operatorname(k,p) \

                          & = p^k(1-p)^
end

</math>

is the binomial distribution where p is a random variable with a beta...
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