If $\theta$ is a ($k$-dimensional) Dirichlet distribution, the sufficient statistics are $\log\theta_i, i = 1,\ldots, k$. It can be shown that if the Dirichlet has parameter $\alpha = (\alpha_1, \ldots, \alpha_k) > 0$, then the expected sufficient statistics are $$\mathbb{E}(\log\theta_i) = \Psi(\alpha_i) - \Psi\left(\sum_{i=1}^k \alpha_i\right)$$ where $\Psi$ is the digamma function.

I would like to compute the space of all realizable mean values $\mathbb{E}(\log\theta)$, i.e. $$\mathcal{M}= \{\mu\in\mathbb{R}^k\,|\,\mu = \mathbb{E_\alpha}(\log\theta) \text{ for some }\alpha\} = \left\{\Psi(\alpha)-\Psi\left(\sum_{i=1}^k\alpha_i\right) \, |\, \alpha \in \mathbb{R}_{>0}^k \right\}$$ Clearly $\mathcal{M} \subseteq \mathbb{R}_{<0}^k$ (because $\Psi$ is increasing), but apart from this however, I am not able to compute the image. Does anyone know what this set is?


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