An Integral Characterization of the Dirichlet Process

Abstract

We give a new integral characterization of the Dirichlet process on a general phase space. To do so, we first prove a characterization of the nonsymmetric Beta distribution via size-biased sampling. Two applications are a new characterization of the Dirichlet distribution and a marked version of a classical characterization of the Poisson–Dirichlet distribution via invariance and independence properties.

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References

  1. 1.

    Dello Schiavo, L., Lytvynov, E.W.: A Mecke-type characterization of the Dirichlet–Ferguson measure. (2017). arXiv:1706.07602

  2. 2.

    Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hoppe, F.M.: Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics. J. Appl. Probab. 23, 1008–1012 (1986)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)

    Google Scholar 

  5. 5.

    Kingman, J.F.C.: Random discrete distributions. J. R. Stat. Soc. Ser. B 37, 1–22 (1975)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kingman, J.F.C.: Poisson Processes. Oxford University Press, Oxford (1993)

    Google Scholar 

  7. 7.

    Last, G., Penrose, M.: Lectures on the Poisson Process. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  8. 8.

    McCloskey, J.W.: A Model for the Distribution of Individuals by Species in an Environment. Unpublished Ph.D. Dissertation Thesis, Michigan State University (1965)

  9. 9.

    Mecke, J.: Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. verw. Gebiete 9, 36–58 (1967)

    Article  Google Scholar 

  10. 10.

    Pakes, A.G.: Characterization by invariance under length-biasing and random scaling. J. Stat. Plan. Inference 63, 285–310 (1997)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Pitman, J.: Some developments of the Blackwell–MacQueen urn scheme. In: Ferguson, T.S., Shapley, L.S., MacQueen, J.B. (eds.), Statistics, Probability and Game Theory. IMS Lecture Notes-Monograph Series, vol. 30, pp. 245–267 (1995)

  12. 12.

    Pitman, J.: Random discrete distributions invariant under size-biased permutation. Adv. Appl. Probab. 28, 525–539 (1996)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Pitman, J.: Combinatorial Stochastic Processes. Ecole de’Éte Probabilités de Saint Flour. Lecture Notes in Mathematics, vol. 1875. Springer, Berlin (2006)

    Google Scholar 

  14. 14.

    Seshadri, V., Wesolowski, J.: Constancy of regressions for beta distributions. Sankhya A 65, 284–291 (2003)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

I wish to thank Lorenzo Dello Schiavo for drawing my attention to the topic of the paper and the referee for making several helpful comments.

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Correspondence to Günter Last.

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Last, G. An Integral Characterization of the Dirichlet Process. J Theor Probab 33, 918–930 (2020). https://doi.org/10.1007/s10959-019-00923-y

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Keywords

  • Dirichlet process
  • Dirichlet distribution
  • Beta distribution
  • Poisson process
  • Mecke equation
  • Poisson–Dirichlet distribution
  • Size-biased sampling

Mathematics Subject Classification (2010)

  • 60G55
  • 60G57