Chinese restaurant process

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]

An equivalent partition process was published a year earlier by Fred Hoppe,^[3] using an "urn scheme" akin to Pólya's urn. In comparison with Hoppe's urn model, the Chinese restaurant process has the advantage that it naturally lends itself to describing random permutations via their cycle structure, in addition to describing random partitions.