Advertisement

Compound Dirichlet Processes

  • Arrigo CoenEmail author
  • Beatriz Godínez-Chaparro
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 301)

Abstract

The compound Poisson process and the Dirichlet process are the pillar structures of renewal theory and Bayesian nonparametric theory, respectively. Both processes have many useful extensions to fulfill the practitioners’ needs to model the particularities of data structures. Accordingly, in this contribution, we join their primal ideas to construct the compound Dirichlet process and the compound Dirichlet process mixture. As a consequence, these new processes have a rich structure to model the time occurrence among events, with also a flexible structure on the arrival variables. These models have a direct Bayesian interpretation of their posterior estimators and are easy to implement. We obtain expressions of posterior distribution, nonconditional distribution, and expected values. In particular, to find these formulas, we analyze sums of random variables with Dirichlet process priors. We assess our approach by applying our model on a real data example of a contagious zoonotic disease.

Keywords

Bayesian nonparametrics Renewal theory Compound poisson process Dirichlet process Random sums 

Notes

Acknowledgements

We thank the editor and two anonymous reviewers for their useful comments which significantly improved the presentation and quality of the paper. The first author is grateful to Prof. Ramsés Mena for the valuable suggestions on an earlier version of the manuscript. This research was partially supported by a DGAPA Postdoctoral Scholarship.

References

  1. 1.
    Aldous, D.J.: Exchangeability and related topics. In: École d’Été de Probabilités de Saint-Flour XIII 1983, pp. 1–198. Springer, Berlin (1985).  https://doi.org/10.1007/BFb0099421Google Scholar
  2. 2.
    Andersson, H., Britton, T.: Stochastic Epidemic Models and Their Statistical Analysis. Lecture Notes in Statistics, vol. 151. Springer, New York, NY (2000)CrossRefGoogle Scholar
  3. 3.
    Blackwell, D., MacQueen, J.B.: Ferguson distributions via polya urn schemes. Ann. Stat. (1973).  https://doi.org/10.1214/aos/1176342372CrossRefzbMATHGoogle Scholar
  4. 4.
    Bladt, M., Nielsen, B.F.: Matrix-Exponential Distributions in Applied Probability. Probability Theory and Stochastic Modelling, vol. 81. Springer, Boston (2017).  https://doi.org/10.1007/978-1-4939-7049-0CrossRefGoogle Scholar
  5. 5.
    Brauer, F., van den Driessche, P., Wu, J. (eds.): Mathematical Epidemiology. Lecture Notes in Mathematics, vol. 1945. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78911-6zbMATHGoogle Scholar
  6. 6.
    Bulla, P., Muliere, P.: Bayesian nonparametric estimation for reinforced markov renewal processes. Stat. Inference Stoch. Process. 10(3), 283–303 (2007).  https://doi.org/10.1007/s11203-006-9000-xMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Coen, A., Gutierrez, L., Mena, R.H.: Modeling failures times with dependent renewal type models via exchangeability. Submited (2018)Google Scholar
  8. 8.
    Coen, A., Mena, R.H.: Ruin probabilities for Bayesian exchangeable claims processes. J. Stat. Plan. Inference 166, 102–115 (2015).  https://doi.org/10.1016/J.JSPI.2015.01.005MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cori, A., Nouvellet, P., Garske, T., Bourhy, H., Nakouné, E., Jombart, T.: A graph-based evidence synthesis approach to detecting outbreak clusters: an application to dog rabies. PLOS Comput. Biol. 14(12), e1006554 (2018).  https://doi.org/10.1371/journal.pcbi.1006554CrossRefGoogle Scholar
  10. 10.
    Doksum, K.: Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2(2), 183–201 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Escobar, M.D.: Estimating normal means with a Dirichlet process prior. J. Am. Stat. Assoc. (1994).  https://doi.org/10.1080/01621459.1994.10476468MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90(430), 577 (1995).  https://doi.org/10.2307/2291069MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. (1995).  https://doi.org/10.1080/01621459.1995.10476550MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ewens, W.J.: The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3(1), 87–112 (1972).  https://doi.org/10.1016/0040-5809(72)90035-4MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ferguson, T.S.: A bayesian analysis of some nonparametric problems. Ann. Stat. 1(2), 209–230 (1973).  https://doi.org/10.1214/aos/1176342360MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ferguson, T.S.: Bayesian density estimation by mixtures of normal distributions. Recent Advances in Statistics, pp. 287–302 (1983).  https://doi.org/10.1016/B978-0-12-589320-6.50018-6CrossRefGoogle Scholar
  17. 17.
    Frees, E.W.: Nonparametric renewal function estimation. Ann. Stat. 14(4), 1366–1378 (1986).  https://doi.org/10.1214/aos/1176350163MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gebizlioglu, O.L., Eryilmaz, S.: The maximum surplus in a finite-time interval for a discrete-time risk model with exchangeable, dependent claim occurrences. Appl. Stoch. Model. Bus. Ind. (2018).  https://doi.org/10.1002/asmb.2415CrossRefGoogle Scholar
  19. 19.
    Ghosal, S., van der Vaart, A.: Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Stat. 35(2), 697–723 (2007).  https://doi.org/10.1214/009053606000001271MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Görür, D., Rasmussen, C.E.: Dirichlet process gaussian mixture models: choice of the base distribution. J. Comput. Sci. Technol. (2010).  https://doi.org/10.1007/s11390-010-9355-8MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ishwaran, H., James, L.F.: Gibbs sampling methods for stick-breaking priors. J. Am. Stat. Assoc. (2001).  https://doi.org/10.1198/016214501750332758MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kingman, J.F.C.: Poisson Processes. Oxford Studies in Probability. Clarendon Press, Oxford (1992)Google Scholar
  23. 23.
    Korwar, R.M., Hollander, M.: Contributions to the theory of dirichlet processes. Ann. Probab. 1(4), 705–711 (1973).  https://doi.org/10.1214/aop/1176996898MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kovalenko, I.N., Pegg, P.A.: Mathematical Theory of Reliability of Time Dependent Systems With Practical Applications. Wiley Series in Probability and Statistics: Applied Probability and Statistics. Wiley, Hoboken (1997)Google Scholar
  25. 25.
    Lenk, P.J.: The logistic normal distribution for bayesian, nonparametric, predictive densities. J. Am. Stat. Assoc. 83(402), 509 (1988).  https://doi.org/10.2307/2288870MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lijoi, A., Mena, R.H., Prünster, I.: Bayesian nonparametric analysis for a generalized dirichlet process prior. Stat. Inference Stoch. Process. 8(3), 283–309 (2005).  https://doi.org/10.1007/s11203-005-6071-zMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lo, A.Y.: On a class of bayesian nonparametric estimates: I. density estimates. Ann. Stat. 12(1), 351–357 (1984).  https://doi.org/10.1214/aos/1176346412MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Maceachern, S.N., Müller, P.: Estimating mixture of dirichlet process models. J. Comput. Graph. Stat. (1998).  https://doi.org/10.1080/10618600.1998.10474772CrossRefGoogle Scholar
  29. 29.
    Nadarajah, S., Li, R.: The exact density of the sum of independent skew normal random variables. J. Comput. Appl. Math. 311, 1–10 (2017).  https://doi.org/10.1016/J.CAM.2016.06.032MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nadarajaha, S., Chanb, S.: The exact distribution of the sum of stable random variables. J. Comput. Appl. Math. 349, 187–196 (2019).  https://doi.org/10.1016/J.CAM.2018.09.044MathSciNetCrossRefGoogle Scholar
  31. 31.
    Neal, R.M.: Markov chain sampling methods for dirichlet process mixture models. J. Comput. Graph. Stat. (2000).  https://doi.org/10.1080/10618600.2000.10474879MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pitman, J.: Some developments of the Blackwell-MacQueen urn scheme. In: Statistics, Probability and Game Theory, pp. 245–267. Institute of Mathematical Statistics, Beachwood (1996).  https://doi.org/10.1214/lnms/1215453576Google Scholar
  33. 33.
    Pitman, J.: Poisson-Kingman partitions. In: Statistics and Science: A Festschrift for Terry Speed, pp. 1–34. Institute of Mathematical Statistics, Beachwood (2003).  https://doi.org/10.1214/lnms/1215091133Google Scholar
  34. 34.
    Prünster, I., Lijoi, A., Regazzini, E.: Distributional results for means of normalized random measures with independent increments. Ann. Stat. 31(2), 560–585 (2003).  https://doi.org/10.1214/aos/1051027881MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. Wiley, Chichester (1999).  https://doi.org/10.1002/9780470317044zbMATHGoogle Scholar
  36. 36.
    Tokdar, S.T.: Posterior consistency of Dirichlet location-scale mixture of normals in density estimation and regression. Sankhy Indian J. Stat. (2003–2007) 68(1), 90–110 (2006)Google Scholar
  37. 37.
    Webber, R.: Communicable Disease Epidemiology and Control: A Global Perspective. Modular Texts, Cabi (2009)CrossRefGoogle Scholar
  38. 38.
    World Health Organization: WHO Expert Consultation on Rabies: Third Report. World Health Organization, Geneva (2018). 92 4 120931 3Google Scholar
  39. 39.
    Xiao, S., Kottas, A., Sansó, B.: Modeling for seasonal marked point processes: an analysis of evolving hurricane occurrences. Ann. Appl. Stat. 9(1), 353–382 (2015).  https://doi.org/10.1214/14-AOAS796MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ypma, R.J.F., Donker, T., van Ballegooijen, W.M., Wallinga, J.: Finding evidence for local transmission of contagious disease in molecular epidemiological datasets. PLoS ONE 8(7), e69875 (2013).  https://doi.org/10.1371/journal.pone.0069875CrossRefGoogle Scholar
  41. 41.
    Zhang, C.H.: Estimation of sums of random variables: examples and information bounds. Ann. Stat. 33(5), 2022–2041 (2005).  https://doi.org/10.1214/009053605000000390MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Facultad de Ciencias, Departamento de MatemáticasUniversidad Nacional Autónoma de MéxicoMexico, CDMXMexico
  2. 2.Departamento de Sistemas Biológicos, División de Ciencias Biológicas y de la SaludUniversidad Autónoma Metropolitana–XochimilcoMexico CityMexico

Personalised recommendations