Pitman–Yor process

In probability theory, a Pitman–Yor process^[1]^[2]^[3]^[4] denoted PY(d, θ, G₀), 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 G₀, 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 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).

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^[5] after Pitman and Yor's review on the subject.^[2] However the process was originally studied in Perman et al^[6]^[7] so technically it perhaps may have been better named the Perman–Pitman–Yor process.

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. However as a name the two-parameter Poisson–Dirichlet process is too long and not very popular. It also emphasizes the strictly decreasing order which is not important in many modeling applications.

References

↑ Ishwaran, H; James, L F (2003). "Generalized weighted Chinese restaurant processes for species sampling mixture models". Statistica Sinica 13: 1211–1211.
↑ 2.0 2.1 Pitman, Jim; Yor, Marc (1997). "The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator". Annals of Probability 25 (2): 855–900. doi:10.1214/aop/1024404422. MR 1434129. Zbl 0880.60076.
↑ Pitman, Jim (2006). Combinatorial Stochastic Processes. Berlin: Springer-Verlag.
↑ Teh, Yee Whye (2006). "A hierarchical Bayesian language model based on Pitman–Yor processes,". Proceedings of the 21st International Conference on Computational Linguistics and the 44th annual meeting of the Association for Computational Linguistics,.
↑ Ishwaran, H.; James, L. (2001). "Gibbs Sampling Methods for Stick-Breaking Priors". Journal of the American Statistical Association.
↑ Perman, M.; Pitman, J.; Yor, M. (1992). "Size-biased sampling of Poisson point processes and excursions". Probability Theory and Related Fields.
↑ Perman, M. (1990). Random Discrete Distributions Derived from Subordinators (Thesis). Department of Statistics, University of California at Berkeley.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itō diffusion Itō process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market SABR volatility Vašíček

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itō integral Itō's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prohorov theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

Pitman–Yor process

Naming conventions

See also

References