In probability theory, a PitmanâÂÂYor process denoted PY(d, ø, G<sub>0</sub>), 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 G<sub>0</sub>, 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 ø > −d and a base distribution G<sub>0</sub> 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 PitmanâÂÂYor process is an example of a Chinese restaurant process, a PoissonâÂÂKingman partition, and of a Gibbs type random partition.
The name "PitmanâÂÂYor process" was coined by Ishwaran and James after Pitman and Yor's review on the subject. However the process was originally studied in Perman et al.
It is also sometimes referred to as the two-parameter PoissonâÂÂDirichlet process, after the two-parameter generalization of the PoissonâÂÂDirichlet distribution which describes the joint distribution of the sizes of the atoms in the random measure, sorted by strictly decreasing order.