Bayesian Analysis

Stochastic Approximations to the Pitman–Yor Process

Julyan Arbel, Pierpaolo De Blasi, and Igor Prünster

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Abstract

In this paper we consider approximations to the popular Pitman–Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error ϵ goes to zero in terms of a polynomially tilted positive stable random variable. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the ϵ-version of the Pitman–Yor process.

Article information

Source
Bayesian Anal., Advance publication (2018), 19 pages.

Dates
First available in Project Euclid: 11 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ba/1560240026

Digital Object Identifier
doi:10.1214/18-BA1127

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

Keywords
stochastic approximation asymptotic distribution Bayesian Nonparametrics Pitman–Yor process random functionals random probability measure stopping rule

Rights
Creative Commons Attribution 4.0 International License.

Citation

Arbel, Julyan; De Blasi, Pierpaolo; Prünster, Igor. Stochastic Approximations to the Pitman–Yor Process. Bayesian Anal., advance publication, 11 June 2019. doi:10.1214/18-BA1127. https://projecteuclid.org/euclid.ba/1560240026


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References

  • Al Labadi, L. and Zarepour, M. (2014). “On simulations from the two-parameter Poisson-Dirichlet process and the normalized Inverse-Gaussian process.” Sankhya, 76-A: 158–176.
    Zentralblatt MATH: 1307.65014
    Digital Object Identifier: doi:10.1007/s13171-013-0033-0
  • Arbel, J., De Blasi, P., and Prünster, I. (2018). “Supplementary Material of “Stochastic Approximations to the Pitman–Yor Process”.” Bayesian Analysis.
  • Bassetti, F., Casarin, R., and Leisen, F. (2014). “Beta-product dependent Pitman–Yor processes for Bayesian inference.” Journal of Econometrics, 180(1): 49–72.
    Zentralblatt MATH: 1298.62148
    Digital Object Identifier: doi:10.1016/j.jeconom.2014.01.007
  • Canale, A., Lijoi, A., Nipoti, B., and Prünster, I. (2017). “On the Pitman-Yor process with spike and slab base measure.” Biometrika, 104: 681–697.
    Zentralblatt MATH: 07072235
    Digital Object Identifier: doi:10.1093/biomet/asx041
  • Caron, F., Neiswanger, W., Wood, F., Doucet, A., and Davy, M. (2017). “Generalized Pólya Urn for Time-Varying Pitman-Yor Processes.” Journal of Machine Learning Research, 18(27): 1–32.
    Zentralblatt MATH: 06781348
  • Devroye, L. (2009). “Random variate generation for exponentially and polynomially tilted stable distributions.” ACM Transactions on Modeling and Computer Simulation (TOMACS), 19(4): Article No. 18.
    Zentralblatt MATH: 1390.65008
    Digital Object Identifier: doi:10.1145/1596519.1596523
  • Favaro, S., Lijoi, A., Mena, R., and Prünster, I. (2009). “Bayesian non-parametric inference for species variety with a two-parameter Poisson–Dirichlet process prior.” Journal of the Royal Statistical Society. Series B, 71: 993–1008.
    Zentralblatt MATH: 1411.62081
    Digital Object Identifier: doi:10.1111/j.1467-9868.2009.00717.x
  • Ferguson, T. S. (1973). “A Bayesian analysis of some nonparametric problems.” Annals of Statistics, 1: 209–230.
    Zentralblatt MATH: 0255.62037
    Digital Object Identifier: doi:10.1214/aos/1176342360
    Project Euclid: euclid.aos/1176342360
  • Gelfand, A. and Kottas, A. (2002). “A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models.” Journal of Computational and Graphical Statistics, 11: 289–305.
  • Ghosal, S. and van der Vaart, A. W. (2017). Foundamentals of Nonparametric Bayesian Inference. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
    Zentralblatt MATH: 1376.62004
  • Gnedin, A. (2004). “The Bernoulli sieve.” Bernoulli, 10: 79–96.
    Zentralblatt MATH: 1044.60005
    Digital Object Identifier: doi:10.3150/bj/1077544604
    Project Euclid: euclid.bj/1077544604
  • Gnedin, A. (2010). “Regeneration in random combinatorial structures.” Probability Surveys, 7: 105–156.
    Zentralblatt MATH: 1204.60028
    Digital Object Identifier: doi:10.1214/10-PS163
  • Gnedin, A., Hansen, B., and Pitman, J. (2007). “Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws.” Probability Surveys, 4: 146–171.
    Zentralblatt MATH: 1189.60050
    Digital Object Identifier: doi:10.1214/07-PS092
  • Gnedin, A., Iksanov, A. M., Pavlo, N., and Uwe, R. (2009). “The Bernoulli sieve revisited.” The Annals of Applied Probability, 19: 1634–1655.
    Zentralblatt MATH: 1178.60019
    Digital Object Identifier: doi:10.1214/08-AAP592
    Project Euclid: euclid.aoap/1248700630
  • Grandell, J. (1997). Mixed Poisson Processes. Monographs on Statistics and Applied Probability. Springer US.
    Zentralblatt MATH: 0922.60005
  • Gut, A. (2013). Probability: a graduate course. Springer texts in statistics. Springer, 2nd ed edition.
    Zentralblatt MATH: 1267.60001
  • Ishwaran, H. and James, L. F. (2001). “Gibbs sampling methods for stick-breaking priors.” Journal of the American Statistical Association, 96: 161–173.
    Zentralblatt MATH: 1014.62006
    Digital Object Identifier: doi:10.1198/016214501750332758
  • Ishwaran, H. and Zarepour, M. (2002). “Exact and approximate sum representations for the Dirichlet Process.” Canadian Journal of Statistics, 30: 269–283.
    Zentralblatt MATH: 1035.60048
    Digital Object Identifier: doi:10.2307/3315951
  • James, L. F., Lijoi, A., and Prünster, I. (2010). “On the posterior distribution of classes of random means.” Bernoulli, 16(1): 155–180.
    Mathematical Reviews (MathSciNet): MR2648753
    Zentralblatt MATH: 1200.62019
    Digital Object Identifier: doi:10.3150/09-BEJ200
    Project Euclid: euclid.bj/1265984707
  • Jara, A. (2007). “Applied Bayesian non- and semi-parametric inference using DPpackage.” Rnews, 7: 17–26.
  • Jara, A., Hanson, T. E., Quintana, F. A., Müller, P., and Rosner, G. L. (2011). “DPpackage: Bayesian Semi- and Nonparametric Modeling in R.” Journal of Statistical Software, 40(5): 1–30.
  • Jara, A., Lesaffre, E., De Iorio, M., and Quintana, F. (2010). “Bayesian semiparametric inference for multivariate doubly-interval-censored data.” Annals of Applied Statistics, 4(4): 2126–2149.
    Zentralblatt MATH: 1220.62023
    Digital Object Identifier: doi:10.1214/10-AOAS368
    Project Euclid: euclid.aoas/1294167813
  • Kingman, J. F. C. (1978). “The representation of partition structures.” Journal of the London Mathematical Society, 18: 374–380.
    Zentralblatt MATH: 0415.92009
    Digital Object Identifier: doi:10.1112/jlms/s2-18.2.374
  • Lijoi, A., Mena, R., and Prünster, I. (2007). “Bayesian nonparametric estimation of the probability of discovering a new species.” Biometrika, 94: 769–786.
    Zentralblatt MATH: 1156.62374
    Digital Object Identifier: doi:10.1093/biomet/asm061
  • Muliere, P. and Tardella, L. (1998). “Approximating distributions of random functionals of Ferguson-Dirichlet priors.” The Canadian Journal of Statistics, 26(2): 283–297.
    Zentralblatt MATH: 0913.62010
    Digital Object Identifier: doi:10.2307/3315511
  • Navarrete, C., Quintana, F. A., and Mueller, P. (2008). “Some issues in nonparametric Bayesian modeling using species sampling models.” Statistical Modelling, 8(1): 3–21.
  • Ni, Y., Müller, P., Zhu, Y., and Ji, Y. (2018). “Heterogeneous reciprocal graphical models.” Biometrics, 74(2): 606–615.
    Zentralblatt MATH: 1414.62466
    Digital Object Identifier: doi:10.1111/biom.12791
  • Perman, M., Pitman, J., and Yor, M. (1992). “Size-biased sampling of Poisson point processes and excursions.” Probability Theory and Related Fields, 92(1): 21–39.
    Zentralblatt MATH: 0741.60037
    Digital Object Identifier: doi:10.1007/BF01205234
  • Pitman, J. (1995). “Exchangeable and partially exchangeable random partitions.” Probability Theory and Related Fields, 102(2): 145–158.
    Zentralblatt MATH: 0821.60047
    Digital Object Identifier: doi:10.1007/BF01213386
  • Pitman, J. (1996). “Some developments of the Blackwell-MacQueen urn scheme.” In Statistics, probability and game theory, volume 30 of IMS Lecture Notes Monogr. Ser., 245–267. Inst. Math. Statist., Hayward, CA.
  • Pitman, J. (2006). Combinatorial stochastic processes. Ecole d’Eté de Probabilités de Saint-Flour XXXII. Lecture Notes in Mathematics N. 1875. Springer, New York.
  • Pitman, J. and Yakubovich, Y. (2017). “Extremes and gaps in sampling from a GEM random discrete distribution.” Electronic Journal of Probability, 22.
    Zentralblatt MATH: 1364.60069
    Digital Object Identifier: doi:10.1214/17-EJP59
  • Pitman, J. and Yor, M. (1997). “The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator.” The Annals of Probability, 25(2): 855–900.
    Zentralblatt MATH: 0880.60076
    Digital Object Identifier: doi:10.1214/aop/1024404422
    Project Euclid: euclid.aop/1024404422
  • Sudderth, E. B. and Jordan, M. I. (2009). “Shared segmentation of natural scenes using dependent Pitman-Yor processes.” In Advances in Neural Information Processing Systems 21, 1585–1592. Curran Associates, Inc.
  • Teh, Y. W. (2006). “A hierarchical Bayesian language model based on Pitman-Yor processes.” In Proc. Coling/ACL, 985–992. Stroudsburg, PA, USA.
  • Tricomi, F. G. and Erdélyi, A. (1951). “The asymptotic expansion of a ratio of gamma functions.” Pacific Journal of Mathematics, 1(1): 133–142.

Supplemental materials

  • Supplementary Material of “Stochastic Approximations to the Pitman–Yor Process”. Algorithm 3 for generating from a polynomially tilted positive stable random variable (in a separate document).
    Digital Object Identifier: doi:10.1214/18-BA1127SUPP
    Supplemental PDF (89 KB)

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