Bayesian Inference: Prior in Chinese Restaurant Process

Question

For the Chinese restaurant process, as used in Dirichlet Process mixture models, we have a prior that data point i belongs to cluster j, where c is an indicator. n represents the total number of data points that are assigned to cluster j, where subscript -i refers to all observations excpet i.

As we take k to infinity, (which represents taking the number of components in the mixture model to infinity), the expression transforms to

when that cluster already exists (which I understand).

For the case that the cluster doesn't already exist, the prior transforms to

This is where my problem lies - how does alpha feature in this expression if (alpha/k) tends to 0 as k tends to infinity?

Thanks in advance

edit: link to paper (pages 3-4) https://www.seas.harvard.edu/courses/cs281/papers/rasmussen-1999a.pdf

Answer 1

This wasn't obvious to me either at first glance. Looking at the first expression, the prior probability that $c_i=c$ where $c$ has no other occupants is $\frac{\alpha/K}{n-1+\alpha}$. Let $J$ be the number of occupied clusters. Then the total prior probability of being in any unoccupied cluster is just $$\frac{\alpha/K}{n-1+\alpha}(K-J)$$ Holding J constant and taking the limit, $$\lim_{K \rightarrow \infty} \frac{\alpha/K}{n-1+\alpha}(K-J)=\frac{\alpha}{n-1+\alpha}\lim_{K\rightarrow \infty}\frac{K-J}{K}=\frac{\alpha}{n-1+\alpha}$$