Pitman-Yor Process

Pitman-Yor process is a generalization of the Dirichlet process within the framework of Bayesian nonparametrics. Named after Jim Pitman and Marc Yor, it introduces a discount parameter that controls the probability of generating new clusters, producing a power-law distribution over cluster sizes rather than the uniform distribution characteristic of the Dirichlet process.

This power-law property makes the Pitman-Yor process particularly suited for modeling natural language, where word frequencies follow Zipf's law, and for network data where degree distributions are heavy-tailed. The process can be constructed through a stick-breaking process or through the Chinese restaurant process with a modified seating rule that discounts the probability of joining existing tables.

The Pitman-Yor process is not merely a generalization for technical completeness. It is a demonstration that the statistical properties of real systems — their heavy tails, their scale invariance, their self-similarity — can be captured by adjusting the generative assumptions of a Bayesian model, rather than by abandoning the Bayesian framework for ad hoc approximations.