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

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

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

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