# Dirichlet-multinomial Distribution - Probability Mass Function - For A Multinomial Distribution Over Category Counts

For A Multinomial Distribution Over Category Counts

For a random vector of category counts, distributed according to a multinomial distribution, the marginal distribution is obtained by integrating out p:

which results in the following explicit formula:

$Pr(mathbf{x}midboldsymbol{alpha})=frac{N!} {prod_{k}left(n_{k}!right)}frac{Gammaleft(Aright)} {Gammaleft(N+Aright)}prod_{k}frac{Gamma(n_{k}+alpha_{k})}{Gamma(alpha_{k})}$

where A is defined as the sum . Note that this differs crucially from the above formula in having an extra term at the front that looks like the factor at the front of a multinomial distribution. Another form for this same compound distribution, written more compactly in terms of the beta function, B, is as follows:

$Pr(mathbf{x}midboldsymbol{alpha})=frac{N Bleft(A,Nright)} {prod_{k:n_k>0} n_k Bleft(alpha_k,n_k right)} .$

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