In
probability theory
Probability theory or probability calculus 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 expre ...
, a Pitman–Yor process
denoted PY(''d'', ''θ'', ''G''
0), is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
whose sample path is a
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
. 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 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 pro ...
. 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
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity var ...
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
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant.
Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Cu ...
, 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
In probability theory, the Chinese restaurant process is a discrete-time stochastic process, analogous to seating customers at tables in a restaurant.
Imagine a restaurant with an infinite number of circular tables, each with infinite capacity. Cu ...
*
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 of pos ...
*
Latent Dirichlet allocation
In natural language processing, latent Dirichlet allocation (LDA) is a Bayesian network (and, therefore, a generative statistical model) for modeling automatically extracted topics in textual corpora. The LDA is an example of a Bayesian topic ...
References
Stochastic processes
Nonparametric Bayesian statistics
Cluster analysis algorithms
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