Skip to main content
SpringerLink
Log in
Book cover

Practical Nonparametric and Semiparametric Bayesian Statistics pp 23–43Cite as

Computational Methods for Mixture of Dirichlet Process Models

  • Chapter

Part of the book series: Lecture Notes in Statistics ((LNS,volume 133))

Abstract

This chapter lays out the basic computational strategies for models based on mixtures of Dirichlet processes. I describe the basic algorithm and give advice on how to improve this algorithm through a collapse of the state space of the Markov chain and through blocking of variates for generation. The computational methods are illustrated with a beta-binomial example and with the bioassay problem. Some advice is given for dealing with models that have little or no conjugacy present.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Antoniak, C.E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems.The Annals of Statistics2, 1152–1174.

    Article  MathSciNet  MATH  Google Scholar 

  • Blackwell, D. and MacQueen, J.B. (1973). Ferguson distributions via Polya urn schemes.The Annals of Statistics1, 353–355.

    Article  MathSciNet  MATH  Google Scholar 

  • Bush, C.A. (1994). Semi-parametric Bayesian Linear Models. Unpublished Ph.D. dissertation, Ohio State University.

    Google Scholar 

  • Bush, C.A. and MacEachern, S.N. (1996). A semi-parametric Bayesian model for randomized block designs.Biometrika83, 275–285.

    Article  MATH  Google Scholar 

  • Dixon, W.J. (1965). The up-and-down method for small samplesJournal of the American Statistical Association60, 967–978.

    Article  MathSciNet  Google Scholar 

  • Doss, H. (1994). Bayesian nonparametric, estimation for incomplete data via successive substitution sampling.The Annals of Statistics22, 1763–1786.

    Article  MathSciNet  MATH  Google Scholar 

  • Doss, H. and Huffer, F. (1998). Monte Carlo methods for Bayesian analysis of survival data using mixtures of Dirichlet priors. Technical Report, Department of Statistics, Ohio State University.

    Google Scholar 

  • Efron, B. and Morris, C. (1975). Data analysis using Stein’s estimator and its generalizations.Journal of the American Statistical Association70, 311–319.

    Article  MATH  Google Scholar 

  • Escobar, M.D. (1994). Estimating normal means with a Dirichlet process prior.Journal of the American Statistical Association89, 268–277.

    Article  MathSciNet  MATH  Google Scholar 

  • Escobar, M.D. and West, M. (1995). Bayesian density estimation and inference using mixtures.Journal of the American Statistical Association90, 577–588.

    Article  MathSciNet  MATH  Google Scholar 

  • Evans, M. and Swartz, T. (1998). Random variable generation using concavity properties of transformed densities.Journal of Computational and Graphical Statisticsto appear.

    Google Scholar 

  • Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems.The Annals of Statistics1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfand, A.E. and Kuo, L. (1991). Nonparametric Bayesian bioassay in-cluding ordered polytomous response.Biometrika78, 657–666.

    Article  MathSciNet  MATH  Google Scholar 

  • Gelfand, A.E. and Smith, A.F.M. (1990). Sampling-based approaches to calculating marginal densities.Journal of the American Statistical Association85, 398–409.

    Article  MathSciNet  MATH  Google Scholar 

  • Gilks, W.R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling.Jounal of Applied Statistics41, 337–348.

    Article  MATH  Google Scholar 

  • Kuo, L. (1983). Bayesian bioassay design.The Annals of Statistics11, 886–895.

    Article  MathSciNet  MATH  Google Scholar 

  • Kuo, L. (1986). Computations of mixtures of Dirichlet processes.SIAM Journal of Scientific and Statistical Computing7, 60–71.

    Article  Google Scholar 

  • Kuo, L. and Smith, A.F.M. (1992). “Bayesian computations in survival models via the Gibbs sampler” inSurvival Analysis: State of the Art(eds: J.P. Klein and P.K. Goel), pp. 11–22.

    Google Scholar 

  • Lindley, D.V. and Smith, A.F.M. (1972). Bayes estimates for the linear model (with discussion).Journal of the Royal Statistical SocietySer. B, 34, 1–42.

    MathSciNet  MATH  Google Scholar 

  • Liu, J.S. (1994). The collapsed Gibbs sampler in Bayesian computations with application to a gene regulation problem.Journal of the American Statistical Association89, 958–966.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, J.S., Wong, W.H. and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes.Biometrika81, 27–40.

    Article  MathSciNet  MATH  Google Scholar 

  • MacEachern, S.N. (1988). Sequential Bayesian bioassay design. Unpublished Ph.D. Dissertation. University of Minnesota.

    Google Scholar 

  • MacEachern, S.N. (1992). Discussion of “Bayesian computations in survival models via the Gibbs sampler” by L. Kuo and A.F.N. Smith, inSurvival Analysis: State of the Art(eds: J.P. Klein and P.K. Goel), pp. 22–23.

    Google Scholar 

  • MacEachern, S.N. (1994). Estimating normal means with a conjugate style Dirichlet process prior.Communications in Statistics: Simulation and Computation23, 727–741.

    Article  MathSciNet  MATH  Google Scholar 

  • MacEachern, S.N., Clyde, M. and Liu, J. (1998). Sequential importance sampling for nonparametric Bayesian models: The next generation.Canadian Journal of Statisticsto appear.

    Google Scholar 

  • MacEachern, S.N. and Müller, P. (1998). Estimating mixtures of Dirichlet process models.Journal of Computational and Graphical Statisticsto appear.

    Google Scholar 

  • Müller, P., Erkanli, A., and West, M. (1996). Bayesian curve fitting using multivariate normal mixtures.Biometrika83, 67–80.

    Article  MathSciNet  MATH  Google Scholar 

  • Newton, MA., Czado, C. and Chappell, R. (1996). Bayesian inference for semiparametric binary regression.Journal of the American Statistical Association91, 142–153.

    Article  MathSciNet  MATH  Google Scholar 

  • Richardson, S. and Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components (with discussion).Journal of the Royal Statistical SocietySer. B, 59, 731–792.

    Article  MathSciNet  MATH  Google Scholar 

  • Walker, S. and Damien, P. (1998). “Sampling methods for Bayesian non-parametric inference involving stochastic processes,” inPractical Non-parametric and Semiparametric Bayesian Statistics(eds: D. Dey, P. Müller, and D. Sinha), New York: Springer-Verlag..

    Google Scholar 

  • West, M., Müller, P. and Escobar, M.D. (1994). “Hierarchical priors and mixture models, with application in regression and density estimation,” inAspects of Uncertainty: A tribute to D.V. Lindley(eds: A.F.M. Smith and P. Freeman), pp. 363–368, New York: Wiley.

    Google Scholar 

Download references

Authors
  1. Steven N. MacEachern
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Statistics, University of Connecticut, Storrs, CT, 06269, USA

    Dipak Dey

  2. ISDS, Duke University, Durham, NC, 27708, USA

    Peter Müller

  3. Department of Mathematics, University of New Hampshire, Durham, NH, 03824, USA

    Debajyoti Sinha

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this chapter

Cite this chapter

MacEachern, S.N. (1998). Computational Methods for Mixture of Dirichlet Process Models. In: Dey, D., Müller, P., Sinha, D. (eds) Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics, vol 133. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1732-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1732-9_2

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98517-6

  • Online ISBN: 978-1-4612-1732-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation