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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.

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    $\begingroup$ 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. $\endgroup$ – Neil G Nov 10 '15 at 3:35

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