Chinese Restaurant Table Distribution

Chinese restaurant table
Parameters	[math]\displaystyle{ \theta \gt 0 }[/math]; [math]\displaystyle{ m \in \{0,1,2,\ldots \} }[/math];
Support	[math]\displaystyle{ L \in \{0,1,2,\ldots,m\} }[/math]
pmf	[math]\displaystyle{ \frac{\Gamma(\theta)}{\Gamma(m+\theta)} \|s(m,\ell)\| \theta^{\ell} }[/math]
Mean	[math]\displaystyle{ \theta (\psi(\theta+m)-\psi(\theta)) }[/math]; (see digamma function)

Short description: probability distribution modeling number of tables in Chinese restaurant process

In probability theory and statistics, the Chinese restaurant table distribution (CRT) is the distribution on the number of tables in the Chinese restaurant process.^[1] It can be understood as the sum of n independent random variables, each with a different Bernoulli distribution:

[math]\displaystyle{ \begin{align} L & = \sum_{n=1}^m b_n \\[4pt] b_n & \sim \operatorname{Bernoulli} \left( \frac \theta {n-1+\theta}\right) \end{align} }[/math]

The probability mass function of L is given by ^[2]

[math]\displaystyle{ f(\ell) = \frac{\Gamma(\theta)}{\Gamma(m+\theta)} |s(m,\ell)| \theta^\ell }[/math]

where s denotes Stirling numbers of the first kind.

References

↑ "Negative Binomial Process Count and Mixture Modeling". https://arxiv.org/abs/1209.3442.
↑ Antoniak, Charles E (1974). "Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems". The Annals of Statistics.

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[1] "Negative Binomial Process Count and Mixture Modeling". https://arxiv.org/abs/1209.3442.

[2] Antoniak, Charles E (1974). "Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems". The Annals of Statistics.

Chinese Restaurant Table Distribution

See also

References