Product of Dirichlet Distributions

The Dirichlet distribution is
\[
D(\vec{x}|\vec{\alpha}) =
\frac{\Gamma\left(\sum_{j=1}^d\alpha_j\right)}{\prod_{j=1}^d\Gamma(\alpha_j)}
\prod_{j=1}^d x_j^{\alpha_j-1}
\]
where \(\Gamma(\cdot)\) is the gamma function and \(d\) is the dimensionality of \(\vec{x}\).

A product of Dirichlet distributions is proportional to another Dirichlet distribution.
\[
\prod_{i=1}^n D(\vec{x}|\vec{\alpha}_i) =
Z\times D(\vec{x}|\vec{\alpha}')
\]
where
\[
\vec{\alpha}' - 1 = \sum_{i=1}^n \left[\vec{\alpha}_i - 1\right]
\]
and
\[
Z = \frac
{\prod_{j=1}^d\Gamma(\alpha'_j)}
{\Gamma\left(\sum_{j=1}^d\alpha'_j\right)}
\ \prod_{i=1}^n\left[ \frac
{\Gamma\left(\sum_{j=1}^d\alpha_{ij}\right)}
{\prod_{j=1}^d\Gamma(\alpha_{ij})} \right]
\]