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[edit]

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]

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.

References[edit]

^ Ishwaran, H; James, L F (2003). "Generalized weighted Chinese restaurant processes for species sampling mixture models". Statistica Sinica. 13: 1211–1235.
^ ^a ^b Pitman, Jim; Yor, Marc (1997). "The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator". Annals of Probability. 25 (2): 855–900. CiteSeerX 10.1.1.69.1273. doi:10.1214/aop/1024404422. MR 1434129. Zbl 0880.60076.
^ Pitman, Jim (2006). Combinatorial Stochastic Processes. Vol. 1875. Berlin: Springer-Verlag. ISBN 9783540309901.
^ 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. 96 (453): 161–173. CiteSeerX 10.1.1.36.2559. doi:10.1198/016214501750332758.
^ Perman, M.; Pitman, J.; Yor, M. (1992). "Size-biased sampling of Poisson point processes and excursions". Probability Theory and Related Fields. 92: 21–39. doi:10.1007/BF01205234.
^ Perman, M. (1990). Random Discrete Distributions Derived from Subordinators (Thesis). Department of Statistics, University of California at Berkeley.

This probability-related article is a stub. You can help Wikipedia by expanding it.

[Ishwaran03-1] Ishwaran, H; James, L F (2003). "Generalized weighted Chinese restaurant processes for species sampling mixture models". Statistica Sinica. 13: 1211–1235.

[PitmanYor1997-2] Pitman, Jim; Yor, Marc (1997). "The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator". Annals of Probability. 25 (2): 855–900. CiteSeerX 10.1.1.69.1273. doi:10.1214/aop/1024404422. MR 1434129. Zbl 0880.60076.

[Pitman2006-3] Pitman, Jim (2006). Combinatorial Stochastic Processes. Vol. 1875. Berlin: Springer-Verlag. ISBN 9783540309901.

[Teh2006-4] 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.

[5] Ishwaran, H.; James, L. (2001). "Gibbs Sampling Methods for Stick-Breaking Priors". Journal of the American Statistical Association. 96 (453): 161–173. CiteSeerX 10.1.1.36.2559. doi:10.1198/016214501750332758.

[PermanPitmanYor1992-6] Perman, M.; Pitman, J.; Yor, M. (1992). "Size-biased sampling of Poisson point processes and excursions". Probability Theory and Related Fields. 92: 21–39. doi:10.1007/BF01205234.

[Perman1990-7] Perman, M. (1990). Random Discrete Distributions Derived from Subordinators (Thesis). Department of Statistics, University of California at Berkeley.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

v t e 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 Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length 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	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
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 Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
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 Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's 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 Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

Naming conventions[edit]

See also[edit]

References[edit]