Pitman–Yor process
Encyclopedia
In probability theory
, a Pitman–Yor process
, denoted PY(d, θ, G0), is a stochastic process
whose sample path is a probability distribution
. A random sample from this process is a finite-dimensional Pitman–Yor distribution, named after Jim Pitman and Marc Yor
. Unfortunately, there is no known analytic form for this distribution.
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).
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a Pitman–Yor process
, denoted PY(d, θ, G0), is a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
whose sample path is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. A random sample from this process is a finite-dimensional Pitman–Yor distribution, named after Jim Pitman and Marc Yor
Marc Yor
Marc Yor is 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 applications to mathematical finance...
. Unfortunately, there is no known analytic form for this distribution.
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
Dirichlet process
In probability theory, a Dirichlet process is a stochastic process that can be thought of as a probability distribution whose domain is itself a random distribution...
. 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).
See also
- Chinese restaurant process
- Dirichlet distribution
- Dirichlet processDirichlet processIn probability theory, a Dirichlet process is a stochastic process that can be thought of as a probability distribution whose domain is itself a random distribution...
- Latent Dirichlet allocationLatent Dirichlet allocationIn statistics, latent Dirichlet allocation is a generative model that allows sets of observations to be explained by unobserved groups that explain why some parts of the data are similar...