# Existence of Indian Buffet Dish Distribution?

The Chinese Restaurant Process has an associated distribution, the Chinese Restaurant Table Distribution, which describes the probability of the number of non-empty tables after N customers have been seated. Does the Indian Buffet process have a similarly associated distribution (perhaps called the Indian Buffet Dish Distribution?) to describe the probability of the number of sampled dishes after N customers have eaten? If so, what are its properties?

Edit: I realized that because the number of sampled dishes is the sum of independent but non-identically distributed Poisson random variables, then can I say the following:

$$\Lambda_N = \sum_{n=1}^N \lambda_n$$

where $$\Lambda_N$$ is the total number of sampled dishes after $$N$$ customers and $$\lambda_n \sim Poisson(\alpha/n)$$ is the number of new dishes sampled by the $$n$$th customer. Because the sum of independent Poissons is itself Poisson, does this mean that

$$\Lambda_N \sim Poisson \Big( \alpha \sum_{n=1}^N \frac{1}{n} \Big)$$

• It might help you the following book "Bayesian Nonparametric Data Analysis", where they discuss in detail the Indian Buffet Process, in Chapter 8. – Fiodor1234 Mar 4 at 16:55