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 conventionsedit
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.
^Ishwaran, H; James, L F (2003). "Generalized weighted Chinese restaurant processes for species sampling mixture models". Statistica Sinica. 13: 1211–1235.
^ abPitman, Jim; Yor, Marc (1997). "The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator". Annals of Probability. 25 (2): 855–900. CiteSeerX10.1.1.69.1273. doi:10.1214/aop/1024404422. MR 1434129. Zbl 0880.60076.
^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. CiteSeerX10.1.1.36.2559. 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.
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pitman, process, probability, theory, denoted, stochastic, process, whose, sample, path, probability, distribution, random, sample, from, this, process, infinite, discrete, probability, distribution, consisting, infinite, atoms, drawn, from, with, weights, dra. In probability theory a Pitman Yor process 1 2 3 4 denoted PY d 8 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 lt 1 a discount parameter a strength parameter 8 gt 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 editThe 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 See also editChinese restaurant process Dirichlet distribution Latent Dirichlet allocationReferences 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 nbsp This probability related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Pitman Yor process amp oldid 1170234820, wikipedia, wiki, book, books, library,
