Skip to main content
SpringerLink
Log in

Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior

  • Published:
Statistical Inference for Stochastic Processes Aims and scope Submit manuscript

Abstract

This paper considers a generalization of the Dirichlet process which is obtained by suitably normalizing superposed independent gamma processes having increasing integer-valued scale parameter. A comprehensive treatment of this random probability measure is provided. We prove results concerning its finite-dimensional distributions, moments, predictive distributions and the distribution of its mean. Most expressions are given in terms of multiple hypergeometric functions, thus highlighting the interplay between Bayesian Nonparametrics and special functions. Finally, a suitable simulation algorithm is applied in order to compute quantities of statistical interest.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • P. Allaart (2003) ArticleTitleMoments of the mean of Dubins-Freedman random probability distributions J. Theor. Probab. 16 471–488 Occurrence Handle10.1023/A:1023582913550 Occurrence Handle1025.60006 Occurrence Handle2004e:60011

    Article  MATH  MathSciNet  Google Scholar 

  • P. Billingsley (2003) Probability and Measure EditionNumber3 Wiley New York

    Google Scholar 

  • L. Bloomer T.P. Hill (2002) ArticleTitleRandom probability measures with given mean and variance J. Theor. Probab. 14 919–937 Occurrence Handle2003h:60072

    MathSciNet  Google Scholar 

  • Bondesson L. Generalized gamma convolutions and related classes of distributions and densities, Lecture Notes in Statistics Vol. 76, Springer, New York, 1992.

  • Cifarelli, D. M. and Regazzini, E. Considerazioni generali sull’impostazione bayesiana di problemi non parametrici. Le medie associative nel contesto del processo aleatorio di Dirichlet Parte II, Rivista di matematica per le scienze economiche e sociali 1 (1979), 95–111.

  • Cifarelli, D. M. and Regazzini, E. Distribution functions of means of a Dirichlet process, Ann. Stat. 18 (1990), 429–442 (Correction in Ann. Statist. 22 (1994), 1633–1634).

  • I. Epifani A. Lijoi I. Prünster (2003) ArticleTitleExponential functionals and means of neutral-to-the-right priors Biometrika 90 791–808 Occurrence Handle10.1093/biomet/90.4.791 Occurrence Handle2004j:60102

    Article  MathSciNet  Google Scholar 

  • M.D. Escobar (1994) ArticleTitleEstimating normal means with a Dirichlet process prior J. Am. Stat. Ass. 89 268–277 Occurrence Handle0791.62039 Occurrence Handle94k:62054

    MATH  MathSciNet  Google Scholar 

  • M.D. Escobar M. West (1995) ArticleTitleBayesian density estimation and inference using mixtures J. Am. Stat. Ass. 90 577–588 Occurrence Handle96d:62054

    MathSciNet  Google Scholar 

  • H. Exton (1976) Multiple hypergeometric functions and applications Ellis Horwood Chichester

    Google Scholar 

  • T.S. Ferguson (1973) ArticleTitleA Bayesian analysis of some nonparametric problems Ann. Stat. 1 209–230 Occurrence Handle0255.62037 Occurrence Handle50 #3441

    MATH  MathSciNet  Google Scholar 

  • D.A. Freedman (1963) ArticleTitleOn the asymptotic behavior of Bayes’ estimates in the discrete case Ann. Math. Stat. 34 1386–1403

    Google Scholar 

  • P. Halmos (1944) ArticleTitleRandom alms Ann. Math. Stat. 15 182–189 Occurrence Handle0060.28302 Occurrence Handle6,5b

    MATH  MathSciNet  Google Scholar 

  • Hjort, N. L. Topics in non–parametric Bayesian statistics, In: P. J. Green et al. (eds), Highly Structured Stochastic Systems, Oxford University Press, Oxford, Oxford Statistical Science Series 27 (2003), 455–478.

  • H. Ishwaran L.F. James (2001) ArticleTitleGibbs sampling methods for stick-breaking priors J. Am. Stat. Ass. 96 IssueID2001 161–173 Occurrence Handle2004d:62018

    MathSciNet  Google Scholar 

  • H. Ishwaran L.F. James (2003) ArticleTitleSome further developments for stick–breaking priors: finite and infinite clustering and classification Sankhya 65 577–592 Occurrence Handle2005a:62147

    MathSciNet  Google Scholar 

  • James, L. F. Poisson Process Partition Calculus with applications to exchangeable models and Bayesian Nonparametrics, Mathematics ArXiv math.PR/0205093 (2002).

  • James, L. F., Lijoi, A. and Prünster, I. Conjugacy as a distinctive feature of the Dirichlet process, Scandinavian Journal of Statistics (2004), to appear.

  • J.F.C. Kingman (1975) ArticleTitleRandom discrete distributions J. Roy Stat. Soc. Ser. B. 37 1–22 Occurrence Handle0331.62019 Occurrence Handle51 #4505

    MATH  MathSciNet  Google Scholar 

  • G. Lauricella (1893) ArticleTitleSulle funzioni ipergeometriche a piùvariabili Rend. Circ. Mat. Palermo 7 111–158 Occurrence Handle25.0756

    MATH  Google Scholar 

  • Lijoi, A., Mena, R. H. and Prünster, I. Hierarchical mixture modelling with normalized inverse Gaussian priors, ICER Working Papers 12/04 (2004).

  • A. Lijoi E. Regazzini (2004) ArticleTitleMeans of a Dirichlet process and multiple hypergeometric functions Ann. Probab. 32 1469–1495 Occurrence Handle2005c:60098

    MathSciNet  Google Scholar 

  • S.N. MacEachern (1994) ArticleTitleEstimating normal means with a conjugate style Dirichlet process prior Commun. Statist. Simulation Comp. 23 727–741 Occurrence Handle0825.62053 Occurrence Handle1293996

    MATH  MathSciNet  Google Scholar 

  • Muliere, P. and Secchi, P. A note on a proper Bayesian bootstrap, Quaderni del Dipartimento di Economia Politica e Metodi Quantitativi 18 (1995), Universtiàdegli Studi di Pavia.

  • Nieto-Barajas, L. E., Prünster, I. and Walker, S. G. Normalized random measures driven by increasing additive processes, Ann Stat. 32 (2004), to appear.

  • S. Petrone A.E. Raftery (1997) ArticleTitleA note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability Stat. Prob. Lett. 36 69–83 Occurrence Handle10.1016/S0167-7152(97)00050-3 Occurrence Handle98k:62067

    Article  MathSciNet  Google Scholar 

  • J. Pitman (1995) ArticleTitleExchangeable and partially exchangeable random partitions Prob. Theory Related Fields 102 145–158 Occurrence Handle10.1007/BF01213386 Occurrence Handle0821.60047 Occurrence Handle96e:60059

    Article  MATH  MathSciNet  Google Scholar 

  • Pitman, J. Some developments of the Blackwell-MacQueen urn scheme, In: T. S. Ferguson et al. (eds), Statistics, probability and game theory. Papers in honor of David Blackwell, Institute of Mathematical Statistics, Hayward, Lecture Notes, Monograph Series Vol 30, 1996, pp. 245–267.

  • Pitman, J. Poisson-Kingman partitions. In: Goldstein, D.R. (ed), Science and Statistics: A Festschrift for Terry Speed, Institute of Mathematical Statistics, Hayward, Lecture Notes, Monograph Series, Vol 40, 2003, pp. 1–35.

  • J. Pitman M. Yor (1997) ArticleTitleThe two-parameter Poisson-Dirichlet distribution derived from a stable subordinator Ann Prob. 25 855–900 Occurrence Handle98f:60147

    MathSciNet  Google Scholar 

  • Prünster, I. Random probability measures derived from increasing additive processes and their application to Bayesian statistics, Ph.d dissertation, Universitàdegli Studi di Pavia, Pavia, 2002.

  • K.B. Oldham J. Spanier (1974) The fractional calculus Academic Press New York

    Google Scholar 

  • Regazzini, E. An example of the interplay between statistics and special functions, In: Tricomi’s ideas and contemporary applied mathematics, Acc. Naz. Lincei e Acc. Scienze Torino, Roma, Atti dei Convegni Lincei Vol. 147, 1998, pp. 303–320.

  • Regazzini, E. Foundations of Bayesian statistics and some theory of Bayesian nonparametric methods, Lecture Notes, Stanford University, Stanford, 2001.

  • E. Regazzini A. Guglielmi G. Di Nunno (2002) ArticleTitleTheory and numerical analysis for exact distribution of functionals of a Dirichlet process Ann. Stat. 30 1376–1411 Occurrence Handle2004h:62054

    MathSciNet  Google Scholar 

  • E. Regazzini A. Lijoi I. Prünster (2003) ArticleTitleDistributional results for means of random measures with independent increments Ann. Stat. 31 560–585

    Google Scholar 

  • K. Sato (1999) Lévy processes and infinitely divisible distributions Cambridge University Press Cambridge

    Google Scholar 

  • J. Sethuraman (1994) ArticleTitleA constructive definition of Dirichlet priors Stat. Sinica 4 639–650 Occurrence Handle0823.62007 Occurrence Handle95m:62058

    MATH  MathSciNet  Google Scholar 

  • S. Skiena (1990) Implementing discrete mathematics: combinatorics and graph theory with Mathematica Addison-Wesley Reading, MA

    Google Scholar 

  • S.G. Walker P. Damien (1998) ArticleTitleA full Bayesian non–parametric analysis involving a neutral to the right process Scand. J Stat. 25 669–680 Occurrence Handle10.1111/1467-9469.00128 Occurrence Handle2000b:62059

    Article  MathSciNet  Google Scholar 

  • S.G. Walker P. Damien P.W. Laud A.F.M. Smith (1999) ArticleTitleBayesian nonparametric inference for random distributions and related functions J. Roy. Stat. Soc. Ser. B. 61 485–527 Occurrence Handle2000g:62123

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Economia Politica e Metodi Quantitativi, Università degli Studi di Pavia, Via San Felice 5, 27100, Pavia, Italy

    Antonio Lijoi

  2. Departimento de Probabilidad y Estadistica, IIMAS-UNAM-México, 04510, Mexico, D.F., Mexico

    Ramsés H. Mena

  3. Dipartimento di Economia Politica e Metodi Quantitativi, Università degli Studi di Pavia, Via San Felice 5, 27100, Pavia, Italy

    Igor Prünster

  4. ICER, Villa Gualino, viale Settimio Severo 63, 10133, Torino, Italy

    Igor Prünster

Authors
  1. Antonio Lijoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ramsés H. Mena
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Igor Prünster
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Lijoi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lijoi, A., Mena, R.H. & Prünster, I. Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior. Stat Infer Stoch Process 8, 283–309 (2005). https://doi.org/10.1007/s11203-005-6071-z

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11203-005-6071-z

Keywords

AMS 200 Mathematics Subject Classifications

Search

Navigation