I have a Chinese Restaurant Process with (unknown) concentration parameter $\theta$. After $n$ customers have been seated, I observe the number of non-empty tables and the number of people at each table. Now I'd like to estimate $\theta$.
How can I estimate $\theta$? e.g., how do I compute the maximum likelihood estimate for $\theta$, or how can I estimate the posterior distribution on $\theta$, given some reasonable/convenient prior?
Crudely, I suppose I could estimate the parameter $\theta$ from the number $t$ of non-empty tables and number $n$ of customers via $\theta \approx t / \log n$, based on a formula for the expected number of non-empty tables as a function of $\theta,n$. However, this doesn't take into account the full data I have, namely, the number of people sitting at each non-empty table.