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 o ...
, a Pitman–Yor process
denoted PY(''d'', ''θ'', ''G''
0), is a
stochastic process whose sample path is a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
. 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.
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 realization (probability), realizations are probability distributions. In other words, a ...
. 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 proportional relative change in the other quantity, independent of the initial size of those quantities: one ...
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, 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 \bolds ...
*
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 exa ...
References
Stochastic processes
Nonparametric Bayesian statistics
Cluster analysis algorithms
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