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I am reading two articles on this and am trying to reach from the exact formula of this (page 19), to the approximate formula of this (section 3.2). Here are the exact and approximated forms using a unified notation:

Exact: $\mathbb{E}(T) = \frac{\theta}{d}\frac{(\theta+d)_{N}}{(\theta)_{N}}-\frac{\theta}{d}$

Approximation: $\mathbb{E}(T)\approx \frac{\Gamma(1+\theta)}{d\Gamma(d+\theta)}N^d$

In these notations, $N$ is total number of elements (aka number of customers in Chinese Restaurant Process terminology), and $T$ is the total number of partitions (aka number of tables in Chinese Restaurant Process terminology), $\Gamma$ is the gamma function, $\theta, \alpha$ are the concentration and discount parameters of Pitman-Yor process, and $(b)_{N}$ is the pochhammer symbol $(b)_{N}=b(b+1)(b+2)...(b+N-1)$.

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  • $\begingroup$ Are we allowed to make any assumptions on the relative sizes of the parameters $\theta$, $d$, and $N$? $\endgroup$ – Gerry Myerson Oct 4 '16 at 6:04
  • $\begingroup$ Yes, $0\leq d < 1$, and $\theta>-d$, and N is typically $N>>d,\theta$, $\endgroup$ – user3639557 Oct 4 '16 at 6:12

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