Multinominal distribution on \([0,\ldots,n-1]\) whose probability of \(i\) is \(p_i\). The logarithm of probability distribution can be written as:
\[
\begin{eqnarray}
\log p(x) &=& \log \left( \sum_{i=0}^{n-1} \delta(i,x) p_x \right)\\
&=& \sum_{i=0}^{n-1} \delta(i,x) \log p_x\\
&=& \left(1 -\sum_{i=1}^{n-1}\delta(i,x)\right)\log p_0 +\sum_{i=1}^{n-1} \delta(i,x) \log p_x\\
&=& \sum_{i=1}^{n-1} \delta(i,x) \log \frac{p_i}{p_0} + \log p_0
\end{eqnarray}
\]
Therefore, by defining
\[
\phi(x) = \delta(i,x),\\
\theta_i = \log \frac{p_i}{p_0},\\
\psi(\theta) = -\log p_0 = \log \left[1+\sum_{i=1}^{n-1}\exp(\theta_i) \right].
\]
the distribution is of exponentail family:
\[
p(x) = \exp \{ \theta \cdot \phi(x) – \psi(\theta) \}
\]