Chinese restaurant table distribution

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

Chinese restaurant table

Parameters	$\theta >0$ $m\in \{0,1,2,\dots \}$
Support	$L\in \{0,1,2,\dots ,m\}$
pmf	$\frac{\mathrm{\Gamma }(\theta )}{\mathrm{\Gamma }(m+\theta )}|s(m,\ell )|{\theta }^{\ell }$
Mean	$\theta (\psi (\theta +m)-\psi (\theta ))$ (see digamma function)

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

$\begin{array}{rl}L& =\sum _{n=1}^m{b}_n\\ b_n& \sim \mathrm{Bernoulli}\left(\frac{\theta }{n-1+\theta }\right)\end{array}$

The probability mass function of L is given by

$f(\ell )=\frac{\mathrm{\Gamma }(\theta )}{\mathrm{\Gamma }(m+\theta )}|s(m,\ell )|{\theta }^{\ell }$

where s denotes Stirling numbers of the first kind.

See also

• Ewens sampling formula