What is the mode of Dirichlet-Multinomial (Polya) distribution?

Question

What is the ML estimate of the parameter $e_i$ for the Dirichlet-Multinomial (Polya) distribution defined below?

$p(\mathbf{x}|\mathbf{e}) = \frac{N!}{\prod_i^d x_i!}\frac{\Gamma(A)}{\Gamma(N+A)}\prod_i^d\frac{\Gamma(x_i+e_i)}{\Gamma(e_i)}$ ,

where $\mathbf{x}$ is a vector of $d$ observations (i.e., how many times we observed the $i^{th}$ value among $d$) and $\mathbf{e}$ is a vector of $d$ prior expectations (i.e., parameters of Dirichlet distribution), with $N=\sum_ix_i$ and $A=\sum_ie_i$.

Answer 1

If you just have a single observation $\mathbf{x}=(x_1,\dots,x_d)$, the likelihood would be maximised by sending each $e_i$ to infinity while keeping $e_i/\sum_{i=1}^d e_i = x_i/\sum_{i=1}^d x_i$ in which case the distribution simplifies to an ordinary multinomial distribution.