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Modelling Heterogeneity With and Without the Dirichlet Process

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  • Peter J. Green
  • Sylvia Richardson

Abstract

We investigate the relationships between Dirichlet process (DP) based models and allocation models for a variable number of components, based on exchangeable distributions. It is shown that the DP partition distribution is a limiting case of a Dirichlet–multinomial allocation model. Comparisons of posterior performance of DP and allocation models are made in the Bayesian paradigm and illustrated in the context of univariate mixture models. It is shown in particular that the unbalancedness of the allocation distribution, present in the prior DP model, persists a posteriori. Exploiting the model connections, a new MCMC sampler for general DP based models is introduced, which uses split/merge moves in a reversible jump framework. Performance of this new sampler relative to that of some traditional samplers for DP processes is then explored.

Suggested Citation

  • Peter J. Green & Sylvia Richardson, 2001. "Modelling Heterogeneity With and Without the Dirichlet Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(2), pages 355-375, June.
  • Handle: RePEc:bla:scjsta:v:28:y:2001:i:2:p:355-375
    DOI: 10.1111/1467-9469.00242
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    File URL: https://doi.org/10.1111/1467-9469.00242
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    1. Villani, Mattias & Kohn, Robert & Nott, David J., 2012. "Generalized smooth finite mixtures," Journal of Econometrics, Elsevier, vol. 171(2), pages 121-133.
    2. Nathan Cunningham & Jim E. Griffin & David L. Wild, 0. "ParticleMDI: particle Monte Carlo methods for the cluster analysis of multiple datasets with applications to cancer subtype identification," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 0, pages 1-22.
    3. Evelina Gabasova & John Reid & Lorenz Wernisch, 2017. "Clusternomics: Integrative context-dependent clustering for heterogeneous datasets," PLOS Computational Biology, Public Library of Science, vol. 13(10), pages 1-29, October.
    4. Sylvia Frühwirth-Schnatter & Gertraud Malsiner-Walli, 2019. "From here to infinity: sparse finite versus Dirichlet process mixtures in model-based clustering," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(1), pages 33-64, March.
    5. Burda, Martin & Harding, Matthew & Hausman, Jerry, 2008. "A Bayesian mixed logit-probit model for multinomial choice," Journal of Econometrics, Elsevier, vol. 147(2), pages 232-246, December.
    6. Li, Mingyang & Meng, Hongdao & Zhang, Qingpeng, 2017. "A nonparametric Bayesian modeling approach for heterogeneous lifetime data with covariates," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 95-104.
    7. Ludkin, Matthew, 2020. "Inference for a generalised stochastic block model with unknown number of blocks and non-conjugate edge models," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    8. Jing Wang, 2010. "Gibbs sampling in DP-based nonlinear mixed effects models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(2), pages 325-340.
    9. Congdon, P., 2007. "Bayesian modelling strategies for spatially varying regression coefficients: A multivariate perspective for multiple outcomes," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2586-2601, February.
    10. Hedibert Freitas Lopes & Peter Müller & Gary L. Rosner, 2003. "Bayesian Meta-analysis for Longitudinal Data Models Using Multivariate Mixture Priors," Biometrics, The International Biometric Society, vol. 59(1), pages 66-75, March.
    11. Alessandra Guglielmi & Francesca Ieva & Anna M. Paganoni & Fabrizio Ruggeri & Jacopo Soriano, 2014. "Semiparametric Bayesian models for clustering and classification in the presence of unbalanced in-hospital survival," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 63(1), pages 25-46, January.
    12. Lau, John W. & So, Mike K.P., 2008. "Bayesian mixture of autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 53(1), pages 38-60, September.
    13. C. Yau & O. Papaspiliopoulos & G. O. Roberts & C. Holmes, 2011. "Bayesian non‐parametric hidden Markov models with applications in genomics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 37-57, January.
    14. Alan P. Ker & Yong Liu, 2017. "Bayesian model averaging of possibly similar nonparametric densities," Computational Statistics, Springer, vol. 32(1), pages 349-365, March.
    15. Nathan Cunningham & Jim E. Griffin & David L. Wild, 2020. "ParticleMDI: particle Monte Carlo methods for the cluster analysis of multiple datasets with applications to cancer subtype identification," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(2), pages 463-484, June.
    16. J. Griffin, 2011. "Bayesian clustering of distributions in stochastic frontier analysis," Journal of Productivity Analysis, Springer, vol. 36(3), pages 275-283, December.
    17. Rosella Castellano & Luisa Scaccia, 2007. "Bayesian inference for Hidden Markov Model," Working Papers 43-2007, Macerata University, Department of Finance and Economic Sciences, revised Oct 2008.

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