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.

Specification

$$\Pr({\mathbf {x}}\mid {\boldsymbol {\alpha }})=\int _{{{\mathbf {p}}}}\Pr({\mathbf {x}}\mid {\mathbf {p}})\Pr({\mathbf {p}}\mid {\boldsymbol {\alpha }}){\textrm {d}}{\mathbf {p}}$$
which results in the following explicit formula:
$$\begin{aligned} \Pr(\mathbf {x} \mid {\boldsymbol {\alpha }}) &{}={\frac {\left(n!\right)\Gamma \left(\alpha _{0}\right)}{\Gamma \left(n+\alpha _{0}\right)}}\prod _{k=1}^{K}{\frac {\Gamma (x_{k}+\alpha _{k})}{\left(x_{k}!\right)\Gamma (\alpha _{k})}}\\ &{}={\frac {nB\left(\alpha _{0},n\right)}{\prod _{k:x_{k}>0}x_{k}B\left(\alpha _{k},x_{k}\right)}} \end{aligned}$$
where $\alpha _{0}$ is defined as the sum ${\displaystyle \alpha _{0}=\sum \alpha _{k}}$ The latter form emphasizes the fact that zero count categories can be ignored in the calculation.

Related distributions

• It reduces to the Categorical distribution as a special case when n = 1: $$\Pr({\mathbf {x}}\mid {\boldsymbol {\alpha }}) = \frac{\alpha_k}{\alpha_0}$$ where $\alpha_k$ is seen as the unnormalized probability of each category.
• It approximates the multinomial distribution arbitrarily well for large α.

Reference

Dirichlet-multinomial Distribution: https://en.wikipedia.org/wiki/Dirichlet-multinomial_distribution