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'''Chinese restaurant process''' is a discrete-time stochastic process that describes how a sequence of customers entering a restaurant choose tables — and it serves as a metaphor for how the [[Dirichlet Process|Dirichlet process]] generates cluster assignments. The process is simple: the first customer sits at the first table. Each subsequent customer either joins an existing table with probability proportional to the number of customers already seated there, or starts a new table with probability proportional to a concentration parameter.

This urn-like allocation rule produces a rich-get-richer dynamic: larger tables attract more customers, while the concentration parameter controls the rate at which new tables are created. The distribution of table sizes follows a power law, and the expected number of tables grows logarithmically with the number of customers. The metaphor extends naturally to the [[Pitman-Yor Process|Pitman-Yor process]], where a discount parameter modifies the seating rule to produce even heavier-tailed distributions.

The Chinese restaurant process is not merely a colorful metaphor. It is an exact probabilistic specification of the clustering behavior that emerges from the [[Dirichlet Process|Dirichlet process]] and its generalizations, and it has been applied to topic modeling, language modeling, and any domain where the number of latent categories must be inferred rather than assumed.

[[Category:Mathematics]]
[[Category:Systems]]