In probability theory, a Pitman–Yor process[1] [2] [3] [4] denoted PY(d, θ, G0), is a stochastic process whose sample path is a probability distribution. A random sample from this process is an infinite discrete probability distribution, consisting of an infinite set of atoms drawn from G0, with weights drawn from a two-parameter Poisson-Dirichlet distribution. The process is named after Jim Pitman and Marc Yor.
The parameters governing the Pitman - Yor process are: 0 ≤ d < 1 a discount parameter, a strength parameter θ > -d and a base distribution G0 over a probability space X. When d = 0, it becomes the Dirichlet process. The discount parameter gives the Pitman - Yor process more flexibility over tail behavior than the Dirichlet process, which has exponential tails. This makes Pitman - Yor process useful for modeling data with power-law tails (e.g., word frequencies in natural language).
The exchangeable random partition induced by the Pitman–Yor process is an example of a Chinese restaurant process, a Poisson–Kingman partition, and of a Gibbs type random partition.
The name "Pitman–Yor process" was coined by Ishwaran and James[5] after Pitman and Yor's review on the subject. However the process was originally studied in Perman et al.[6] [7]
It is also sometimes referred to as the two-parameter Poisson–Dirichlet process, after the two-parameter generalization of the Poisson–Dirichlet distribution which describes the joint distribution of the sizes of the atoms in the random measure, sorted by strictly decreasing order.