# Is the Dirichlet compound multinomial (DCM) distribution in the exponential family?

It occurs to me that the DCM (a.k.a multivariate Pòlya) distribution can be written in the exponential form when the number of draws, $n_1+n_2+\ldots+n_k=N$, is known: $$p(n|a) = exp \left(tr \left( \theta(a)^\top T(n) \right) - b(a) \right)$$ where, for $i \in \{0,1,\ldots,N-1\}$ and $j \in \{1,2,\ldots,k\}$, \begin{align} \theta_{ij} &= \ln \frac{a_j+i}{1+i} \\ T_{ij} &= \begin{cases} 1 & i < n_j \\ 0 & i \geq n_j. \end{cases} \end{align} The identity needed in the short derivation is, for non-negative integer $k$: $$\ln \frac{\Gamma(z+k)}{\Gamma(z)} = \sum_{i=0}^{k-1} \ln(z+i).$$ The log-partition function is similarly found to be $$b(a) = \sum_{i=0}^{N-1} \ln \frac{A + i}{1 + i},$$ where $a_1+a_2+\ldots+a_k=A$.

Although the dimension of the parameter and statistic are larger than the dimension of the random vector, I don't believe that violates the definition of exponential families. If wrong though, I would like to be corrected. The use of the matrix trace is just a convenient shortcut to the inner product of $N \times k$ vectors. So, why isn't the DCM (and thus the beta-binomial) typically included in the exponential family? I grant you, the statistic $T$ is kind of silly.

• There's nothing wrong with having more parameters and sufficient statistics than the random vector. The Gaussian distribution has two parameters and one element in its random vector. – Neil G Nov 10 '15 at 3:35