Pitman–Yor process
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, a Pitman–Yor process denoted PY(''d'', ''θ'', ''G''0), 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''0, with weights drawn from a two-parameter Poisson–Dirichlet distribution. The process is named after
Jim Pitman Jim or JIM may refer to: * Jim (given name), a given name * Jim, a diminutive form of the given name James (given name), James * Jim, a short form of the given name Jimmy (given name), Jimmy * OPCW-UN Joint Investigative Mechanism * Jim (comics), ...
and
Marc Yor Marc Yor (24 July 1949 – 9 January 2014) was a French mathematician well known for his work on stochastic processes, especially properties of semimartingales, Brownian motion and other Lévy processes, the Bessel processes, and their applicat ...
. The parameters governing the Pitman–Yor process are: 0 ≤ ''d'' < 1 a discount parameter, a strength parameter ''θ'' > −''d'' and a base distribution ''G''0 over a probability space  ''X''. When ''d'' = 0, it becomes the
Dirichlet process In probability theory, Dirichlet processes (after the distribution associated with Peter Gustav Lejeune Dirichlet) are a family of stochastic processes whose realizations are probability distributions. In other words, a Dirichlet process is a p ...
. 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 Poisson–Kingman partition, and of a Gibbs type random partition.


Naming conventions

The name "Pitman–Yor 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 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 In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. ...
, sorted by strictly decreasing order.


See also

* Chinese restaurant process *
Dirichlet distribution In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \bold ...
*
Latent Dirichlet allocation In natural language processing, Latent Dirichlet Allocation (LDA) is a generative statistical model that explains a set of observations through unobserved groups, and each group explains why some parts of the data are similar. The LDA is an ex ...


References

Stochastic processes Nonparametric Bayesian statistics Cluster analysis algorithms {{probability-stub