Multivariate Pólya distribution

Dirichlet-Multinomial
Parameters	$n>0$ number of trials (integer) $\alpha _{1},\ldots ,\alpha _{K}>0$
Support	$x_{i}\in \{0,\dots ,n\}$ $\Sigma x_{i}=n\!$
pmf	${\frac {\left(n!\right)\Gamma \left(\sum \alpha _{k}\right)}{\Gamma \left(n+\sum \alpha _{k}\right)}}\prod _{k=1}^{K}{\frac {\Gamma (x_{k}+\alpha _{k})}{\left(x_{k}!\right)\Gamma (\alpha _{k})}}$
Mean	$E(X_{i})=n{\frac {\alpha _{i}}{\sum \alpha _{k}}}$
Variance	$Var(X_{i})=n{\frac {\alpha _{i}}{\sum \alpha _{k}}}\left(1-{\frac {\alpha _{i}}{\sum \alpha _{k}}}\right)\left({\frac {n+\sum \alpha _{k}}{1+\sum \alpha _{k}}}\right)$ $\textstyle {\mathrm {Cov} }(X_{i},X_{j})=-n{\frac {\alpha _{i}\alpha _{j}}{(\sum \alpha _{k})^{2}}}\left({\frac {n+\sum \alpha _{k}}{1+\sum \alpha _{k}}}\right)~~(i\neq j)$

In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector ${\boldsymbol {\alpha }}$ , and an observation drawn from a multinomial distribution with probability vector p and number of trials N. The compounding corresponds to a Polya urn scheme. It is frequently encountered in Bayesian statistics, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.

It reduces to the Categorical distribution as a special case when n = 1. It also approximates the multinomial distribution arbitrarily well for large α. The Dirichlet-multinomial is a multivariate extension of the Beta-binomial distribution, as the multinomial and Dirichlet distributions are multivariate versions of the binomial distribution and beta distributions, respectively.