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Consider a multinomial distribution in 4 elements. Let’s write this in terms of a n exponential family. Consider:
\begin{equation} \begin{cases} T\qty(1) = \mqty(1 & 0 & 0) \\ T\qty(2) = \mqty(0&1&0) \\ T\qty(3) = \mqty(0&0&1) \end{cases} \end{equation}
And, given \(\phi_{1},\phi_{2},\phi_{3}\):
\begin{equation} p\qty(y) = \phi_{1}^{T\qty(y)_{1}} \phi_{2}^{T\qty(y)_{2}}\phi_{3}^{T\qty(y)_{3}}\phi_{4}^{1-\qty(T\qty(y)_{1}+T\qty(y)_{2}+T\qty(y)_{3})} \end{equation}
Taking the \(\exp \log \qty(^{})\) of the above, we obtain:
\begin{equation} p\qty(y) = \exp \qty(T\qty(y)_{1}\log \frac{\phi_{1}}{\phi_{4}} + T\qty(y)_{2} \log \frac{\phi_{2}}{\phi_{4}} + T\qty(y)_{3} \log \frac{\phi_{3}}{\phi_{4}} + \log\qty(\phi_{4})) \end{equation}
Which we can now rewrite in the standard form of an exponential family, for which \(b\qty(y) = 1\) and then:
\begin{equation} \eta = \mqty(\log \frac{\phi_{1}}{\phi_{4}} \\ \log \frac{\phi_{2}}{\phi_{4}} \\ \log \frac{\phi_{3}}{\phi_{4}}) \end{equation}