Is the Generalized Dirichlet distribution an exponential family? If so, what is its log-normalizer, sufficient statistics, and carrier measure?
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Yes, it is an exponential family.
It has sufficient statistics: $\{\log(x_i)\}_i$ and $\{\log(1 - \sum_{j \le i}x_j)\}_i$.
It has zero carrier measure.
Its log-normalizer is $\sum_i B(\alpha_i, \beta_i)$ where the natural parameters that correspond to the above sufficient statistics are $\{\alpha_i-1\}_i$ and $\{\gamma_i\}_i$ with $\gamma_j=\beta_j-\alpha_{j+1}-\beta_{j+1}$.
After some work, I was able to implement the distribution here.