In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant. Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Customer 1 sits at the first table. The next customer either sits at the same table as customer 1, or the next table. This continues, with each customer choosing to either sit at an occupied table with a probability proportional to the number of customers already there (i.e., they are more likely to sit at a table with many customers than few), or an unoccupied table. At time n, the n customers have been partitioned among m ≤ n tables (or blocks of the partition). The results of this process are exchangeable, meaning the order in which the customers sit does not affect the probability of the final distribution. This property greatly simplifies a number of problems in population genetics, linguistic analysis, and image recognition.

The restaurant analogy first appeared in a 1985 write-up by David Aldous,^[1] where it was attributed to Jim Pitman (who additionally credits Lester Dubins).^[2]

Formal definition

For any positive integer ${\displaystyle n}$, let ${\displaystyle {\mathcal {P}}_{n}}$ denote the set of all partitions of the set ${\displaystyle \{1,2,3,...,n\}\triangleq [n]}$. The Chinese restaurant process takes values in the infinite Cartesian product ${\displaystyle \prod _{n\geq 1}{\mathcal {P}}_{n}}$.

The value of the process at time ${\displaystyle n}$ is a partition ${\displaystyle B_{n}}$ of the set ${\displaystyle [n]}$, whose probability distribution is determined as follows. At time ${\displaystyle n=1}$, the trivial partition ${\displaystyle B_{1}=\{\{1\}\}}$ is obtained (with probability one). At time ${\displaystyle n+1}$ the element "${\displaystyle n+1}$" is either:

1. added to one of the blocks of the partition ${\displaystyle B_{n}}$, where each block is chosen with probability ${\displaystyle |b|/(n+1)}$ where ${\displaystyle |b|}$ is the size of the block (i.e. number of elements), or
2. added to the partition ${\displaystyle B_{n}}$ as a new singleton block, with probability ${\displaystyle 1/(n+1)}$.

The random partition so generated has some special properties. It is exchangeable in the sense that relabeling ${\displaystyle \{1,...,n\}}$ does not change the distribution of the partition, and it is consistent in the sense that the law of the partition of ${\displaystyle [n-1]}$ obtained by removing the element ${\displaystyle n}$ from the random partition ${\displaystyle B_{n}}$ is the same as the law of the random partition ${\displaystyle B_{n-1}}$.

The probability assigned to any particular partition (ignoring the order in which customers sit around any particular table) is

${\displaystyle \Pr(B_{n}=B)={\frac {\prod _{b\in B}(|b|-1)!}{n!}},\qquad B\in {\mathcal {P}}_{n}}$

where ${\displaystyle b}$ is a block in the partition ${\displaystyle B}$ and ${\displaystyle |b|}$ is the size of ${\displaystyle b}$.

The definition can be generalized by introducing a parameter ${\displaystyle \theta >0}$ which modifies the probability of the new customer sitting at a new table to ${\displaystyle {\frac {\theta }{n+\theta }}}$ and correspondingly modifies the probability of them sitting at a table of size ${\displaystyle |b|}$ to ${\displaystyle {\frac {|b|}{n+\theta }}}$. The vanilla process introduced above can be recovered by setting ${\displaystyle \theta =1}$. Intuitively, ${\displaystyle \theta }$ can be interpreted as the effective number of customers sitting at the first empty table.

Distribution of the number of tables

Chinese restaurant table

Parameters	${\displaystyle \theta >0}$ ${\displaystyle n\in \{1,2,\ldots \}}$
Support	${\displaystyle k\in \{1,2,\ldots ,n\}}$
PMF	${\displaystyle {\frac {\Gamma (\theta )}{\Gamma (n+\theta )}}|s(n,k)|\theta ^{k}}$
Mean	${\displaystyle \theta (\psi (\theta +n)-\psi (\theta ))}$ (see digamma function)

The Chinese restaurant table distribution (CRT) is the probability distribution on the number of tables in the Chinese restaurant process.^[4] It can be understood as the sum of ${\displaystyle n}$ independent Bernoulli random variables, each with a different parameter:

${\displaystyle {\begin{aligned}K&=\sum _{i=1}^{n}b_{i}\\[4pt]b_{i}&\sim \operatorname {Bernoulli} \left({\frac {\theta }{i-1+\theta }}\right)\end{aligned}}}$

The probability mass function of ${\displaystyle K}$ is given by ^[5]

${\displaystyle f(k)={\frac {\Gamma (\theta )}{\Gamma (n+\theta )}}|s(n,k)|\theta ^{k},\quad k=1,\dots ,n,}$

where ${\displaystyle s}$ denotes Stirling numbers of the first kind.

Generalization

This construction can be generalized to a model with two parameters, ${\displaystyle \theta }$ & ${\displaystyle \alpha }$,^[2][6] commonly called the strength (or concentration) and discount parameters respectively. At time ${\displaystyle n+1}$, the next customer to arrive finds ${\displaystyle |B|}$ occupied tables and decides to sit at an empty table with probability

${\displaystyle {\frac {\theta +|B|\alpha }{n+\theta }},}$

or at an occupied table ${\displaystyle b}$ of size ${\displaystyle |b|}$ with probability

${\displaystyle {\frac {|b|-\alpha }{n+\theta }}.}$

In order for the construction to define a valid probability measure it is necessary to suppose that either ${\displaystyle \alpha <0}$ and ${\displaystyle \theta =-L\alpha }$ for some ${\displaystyle L\in \{1,2,,...\}}$; or that ${\displaystyle 0\leq \alpha <1}$ and ${\displaystyle \theta >-\alpha }$.

Under this model the probability assigned to any particular partition ${\displaystyle B}$ of ${\displaystyle [n]}$, in terms of the Pochhammer k-symbol, is

${\displaystyle \Pr(B_{n}=B)={\frac {(\theta +\alpha )_{|B|-1,\alpha }}{(\theta +1)_{n-1,1}}}\prod _{b\in B}(1-\alpha )_{|b|-1,1}}$

where, by convention, ${\displaystyle (a)_{0,c}=1}$, and for ${\displaystyle b>0}$

${\displaystyle (a)_{b,c}=\prod _{i=0}^{b-1}(a+ic)={\begin{cases}a^{b}&{\text{if }}c=0,\\\\{\dfrac {c^{b}\,\Gamma (a/c+b)}{\Gamma (a/c)}}&{\text{otherwise}}.\end{cases}}}$

Thus, for the case when ${\displaystyle \theta >0}$ the partition probability can be expressed in terms of the Gamma function as

${\displaystyle \Pr(B_{n}=B)={\frac {\Gamma (\theta )}{\Gamma (\theta +n)}}{\dfrac {\alpha ^{|B|}\,\Gamma (\theta /\alpha +|B|)}{\Gamma (\theta /\alpha )}}\prod _{b\in B}{\dfrac {\Gamma (|b|-\alpha )}{\Gamma (1-\alpha )}}.}$

In the one-parameter case, where ${\displaystyle \alpha }$ is zero, this simplifies to

${\displaystyle \Pr(B_{n}=B)={\frac {\Gamma (\theta )\,\theta ^{|B|}}{\Gamma (\theta +n)}}\prod _{b\in B}\Gamma (|b|).}$

Or, when ${\displaystyle \theta }$ is zero,

${\displaystyle \Pr(B_{n}=B)={\frac {\alpha ^{|B|-1}\,\Gamma (|B|)}{\Gamma (n)}}\prod _{b\in B}{\frac {\Gamma (|b|-\alpha )}{\Gamma (1-\alpha )}}.}$

As before, the probability assigned to any particular partition depends only on the block sizes, so as before the random partition is exchangeable in the sense described above. The consistency property still holds, as before, by construction.

If ${\displaystyle \alpha =0}$, the probability distribution of the random partition of the integer ${\displaystyle n}$ thus generated is the Ewens distribution with parameter ${\displaystyle \theta }$, used in population genetics and the unified neutral theory of biodiversity.

Animation of a Chinese restaurant process with scaling parameter ${\displaystyle \theta =0.5,\ \alpha =0}$. Tables are hidden once the customers of a table can not be displayed anymore; however, every table has infinitely many seats. (Recording of an interactive animation.^[7])

Applications

The Chinese restaurant process is closely connected to Dirichlet processes and Pólya's urn scheme, and therefore useful in applications of Bayesian statistics including nonparametric Bayesian methods. The Generalized Chinese Restaurant Process is closely related to Pitman–Yor process. These processes have been used in many applications, including modeling text, clustering biological microarray data,^[10] biodiversity modelling, and image reconstruction ^[11][12]

See also

• Ewens sampling formula
• Preferential attachment
• Hilbert's paradox of the Grand Hotel

References

External links

• Introduction to the Dirichlet Distribution and Related Processes by Frigyik, Kapila and Gupta
• A talk by Michael I. Jordan on the CRP:
  • http://videolectures.net/icml05_jordan_dpcrp/