Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k \alpha_k)}{\sum_k \Gamma(\alpha_k)}\cdot \prod_k \theta_k^{\alpha_k-1}$$

What is the maximum likelihood estimate (MLE) w.r.t. $\alpha$ ? that is, solve

$$\arg\max_\alpha p(\theta \mid \alpha)$$

This problem is a simplified case of a problem described in Thomas Minka's paper "Estimating a Dirichlet Distribution". In his problem, more than one $\theta$ is observed.

I believe it should be easy to show the maximum value is $\infty$, by setting $\alpha := c\cdot\theta$ and $c = \infty$, but am unable to show this. Can anyone derive a proof, or counter example?


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