Dirichlet process vs. Pitman-Yor and power-law - how do they relate?

Question

The Dirichlet process has, on expectation, $\alpha \log n$ "restaurants", when borrowing the Chinese restaurant process terminology, with $\alpha$ being the concentration parameter.

Pitman-Yor, on the other hand, has $\alpha n^d$ "tables", with $\alpha$ being the concentration parameter, and $d$ being the discount parameter.

How does that relate to power-law? I understand PY has power-law behavior, but DP does not.

From Wikipedia:

"The parameters governing the Pitman–Yor process are: 0 ≤ d < 1 a discount parameter, a strength parameter θ > −d and a base distribution G0 over a probability space X. When d = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman–Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman–Yor process useful for modeling data with power-law tails "

This, unfortunately, is not clear enough for me.

Answer 1

Hopefully someone can give you a better answer, but I think the main point is this: the probabilities associated to each cluster decay, on average, exponentially for the Dirichlet process. Consider the stick breaking construction, where we let $\beta'_k \sim \mbox{Beta}(1, \alpha)$ and $\beta_k = \beta'_k \prod_{j < k} (1 - \beta'_j)$ where $\beta_k$ is the probability of landing in cluster $k$. Pitman-Yor processes have similar stick breaking constructions.

What happens to the probability of landing in cluster $k$ as $k \to \infty$? For the Dirichlet process this goes down exponentially fast - as the table I'm asking about becomes more and more obscure (say) my probability of ending up there goes down exponentially. If $\alpha = 1$, for example, the class probabilities might go down like $\frac 1 2$, $\frac 1 4$, $\frac 1 8$, .... (this would occur if each $\beta'_k$ were equal to its expectation). For the Pitman-Yor process, however, the probability does not go down exponentially, but instead like a power law. If we interpret the tables as being topics in a topic model, and we know a priori that distributions of topics have power law tails, then we would like the probability of me ending up at obscure latent topics to go down like a power law, so the Pitman-Yor might be more appropriate.