Bayesian Hierarchical Mixture Models

  • Leonardo BottoloEmail author
  • Petros Dellaportas
Conference paper
Part of the Abel Symposia book series (ABEL, volume 11)


When massive streams of data are collected, it is usually the case that different information sources contribute to different levels of knowledge, or inferences about subgroups may suffer from small or inadequate sample size. In these cases, Bayesian hierarchical models have been proven to be valuable, or even necessary, modelling tools that provide the required multi-level modelling structure to deal with the statistical inferential procedure. We investigate the need to generalize the inherent assumption of exchangeability which routinely accompanies these models. By modelling the second-stage parameters of a Bayesian hierarchical model as a finite mixture of normals with unknown number of components, we allow for parameter partitions so that exchangeability is assumed within each partition. This more general model formulation allows better understanding of the data generating mechanism and provides better parameter estimates and forecasts. We discuss choices of prior densities and MCMC implementation in problems in actuarial science, finance and genetics.


Finite Mixture Dirichlet Process Bayesian Hierarchical Model Prior Density Prior Specification 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors acknowledge financial support from the Royal Society (International Exchanges grant IE110977). The second author acknowledges financial support from the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) through the research funding program ARISTEIA-LIKEJUMPS.


  1. 1.
    Albert, J., Chib, S.: Bayesian tests and model diagnostics in conditionally independent hierarchical models. J. Am. Stat. Assoc. 92(439), 916–925 (1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bhattacharjee, A., Richards, W.G., Staunton, J., Li, C., Monti, S., Vasa, P., Ladd, C., Beheshti, J., Bueno, R., Gillette, M., et al.: Classification of human lung carcinomas by mrna expression profiling reveals distinct adenocarcinoma subclasses. Proc. Natl. Acad. Sci. USA 98(24), 13790–13795 (2001)CrossRefGoogle Scholar
  3. 3.
    Bottolo, L., Consonni, G.: Bayesian clustering of gene expression microarray data for subgroup identification. In: Atti della XLII Riunione Scientifica della Societa’ Italiana di Statistica, pp. 187–198. CLEUP, Padova (2004)Google Scholar
  4. 4.
    Bottolo, L., Dellaportas, P.: Bayesian hierarchical mixture models for financial time series (2015, in preparation)Google Scholar
  5. 5.
    Bottolo, L., Consonni, G., Dellaportas, P., Lijoi, A.: Bayesian analysis of extreme values by mixture modeling. Extremes 6(1), 25–47 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Consonni, G., Veronese, P.: A Bayesian method for combining results from several binomial experiments. J. Am. Stat. Assoc. 90(431), 935–944 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Dawid, A.P.: Exchangeability and its ramifications. In: Damien, P., Dellaportas, P., Polson, N.G., Stephens, D.A. (eds.) Bayesian Theory and Applications, pp. 19–30. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
  8. 8.
    Escobar, M.D., West, M.: Bayesian density estimation and inference using mixtures. J. Am. Stat. Assoc. 90(430), 577–588 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Garrett, E.S., Parmigiani, G.: POE: statistical methods for qualitative analysis of gene expression. In: The Analysis of Gene Expression Data, pp. 362–387. Springer, Berlin (2003)Google Scholar
  10. 10.
    Gelfand, A.E., Ghosh, S.: Hierarchical modelling. In: Damien, P., Dellaportas, P., Polson, N.G., Stephens, D.A. (eds.) Bayesian Theory and Applications, pp. 33–49. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
  11. 11.
    George, E.I., Liang, F., Xu, X.: From minimax shrinkage estimation to minimax shrinkage prediction. Stat. Sci. 27(1), 82–94 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Green, P.J., Richardson, S.: Modelling heterogeneity with and without the Dirichlet process. Scand. J. Stat. 28(2), 355–375 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Ibrahim, J.G., Chen, M.-H.: Power prior distributions for regression models. Stat. Sci. 15(1), 46–60 (2000)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lee, H., Li, J.: Variable selection for clustering by separability based on ridgelines. J. Comput. Graph. Stat. 21(2), 315–337 (2012)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Lin, L., Chan, C., West, M.: Discriminative variable subsets in bayesian classification with mixture models, with application in flow cytometry studies. Biostatistics (2015, in press)Google Scholar
  16. 16.
    Lindley, D.V., Smith, A.F.: Bayes estimates for the linear model. J. R. Stat. Soc. B 34(1), 1–41 (1972)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Martino, L., Read, J.: On the flexibility of the design of multiple try Metropolis schemes. Comput. Stat. 28(6), 2797–2823 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Nobile, A., Green, P.J.: Bayesian analysis of factorial experiments by mixture modelling. Biometrika 87(1), 15–35 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Parmigiani, G., Garrett, E.S., Anbazhagan, R., Gabrielson, E.: A statistical framework for expression-based molecular classification in cancer. J. R. Stat. Soc. B 64(4), 717–736 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Richardson, S., Green, P.J.: On Bayesian analysis of mixtures with an unknown number of components (with discussion). J. R. Stat. Soc. B 59(4), 731–792 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Smith, R.L.: Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Stat. Sci. 4(4), 367–377 (1989)CrossRefzbMATHGoogle Scholar
  23. 23.
    Smith, R.L., Goodman, D.: Bayesian risk analysis. In: Embrechts, P. (ed.) Extremes and Integrated Risk Management, pp. 235–251. Risk Books, London (2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Department of StatisticsAthens University of Economics and BusinessAthensGreece

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