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Controlling the reinforcement in Bayesian non-parametric mixture models

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  • Antonio Lijoi
  • Ramsés H. Mena
  • Igor Prünster

Abstract

The paper deals with the problem of determining the number of components in a mixture model. We take a Bayesian non-parametric approach and adopt a hierarchical model with a suitable non-parametric prior for the latent structure. A commonly used model for such a problem is the mixture of Dirichlet process model. Here, we replace the Dirichlet process with a more general non-parametric prior obtained from a generalized gamma process. The basic feature of this model is that it yields a partition structure for the latent variables which is of Gibbs type. This relates to the well-known (exchangeable) product partition models. If compared with the usual mixture of Dirichlet process model the advantage of the generalization that we are examining relies on the availability of an additional parameter "σ" belonging to the interval (0,1): it is shown that such a parameter greatly influences the clustering behaviour of the model. A value of "σ" that is close to 1 generates a large number of clusters, most of which are of small size. Then, a reinforcement mechanism which is driven by "σ" acts on the mass allocation by penalizing clusters of small size and favouring those few groups containing a large number of elements. These features turn out to be very useful in the context of mixture modelling. Since it is difficult to specify "a priori" the reinforcement rate, it is reasonable to specify a prior for "σ". Hence, the strength of the reinforcement mechanism is controlled by the data. Copyright 2007 Royal Statistical Society.

Suggested Citation

  • Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2007. "Controlling the reinforcement in Bayesian non-parametric mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(4), pages 715-740.
  • Handle: RePEc:bla:jorssb:v:69:y:2007:i:4:p:715-740
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9868.2007.00609.x
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    References listed on IDEAS

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    1. Lancelot F. James & Antonio Lijoi & Igor Pruenster, 2005. "Bayesian Inference via Classes of Normalized Random Measures," ICER Working Papers - Applied Mathematics Series 5-2005, ICER - International Centre for Economic Research.
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    Cited by:

    1. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2009. "Posterior Analysis for Normalized Random Measures with Independent Increments," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 76-97.
    2. Stefano Favaro & Antonio Lijoi & Igor Prünster, 2012. "A New Estimator of the Discovery Probability," Biometrics, The International Biometric Society, vol. 68(4), pages 1188-1196, December.
    3. Pierpaolo De Blasi & Stefano Favaro & Antonio Lijoi & Ramsés H. Mena & Igor Prünster & Mattteo Ruggiero, 2013. "Are Gibbs-type priors the most natural generalization of the Dirichlet process?," DEM Working Papers Series 054, University of Pavia, Department of Economics and Management.
    4. Raffaele Argiento & Alessandra Guglielmi & Antonio Pievatolo, 2014. "Estimation, prediction and interpretation of NGG random effects models: an application to Kevlar fibre failure times," Statistical Papers, Springer, vol. 55(3), pages 805-826, August.
    5. J. E. Griffin & M. Kolossiatis & M. F. J. Steel, 2013. "Comparing distributions by using dependent normalized random-measure mixtures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 499-529, June.
    6. Antonio Lijoi & Bernardo Nipoti & Igor Prünster, 2013. "Dependent mixture models: clustering and borrowing information," DEM Working Papers Series 046, University of Pavia, Department of Economics and Management.
    7. Kolossiatis, M. & Griffin, J.E. & Steel, M.F.J., 2011. "Modeling overdispersion with the normalized tempered stable distribution," Computational Statistics & Data Analysis, Elsevier, vol. 55(7), pages 2288-2301, July.
    8. Lijoi, Antonio & Nipoti, Bernardo & Prünster, Igor, 2014. "Dependent mixture models: Clustering and borrowing information," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 417-433.
    9. Cesari, Oriana & Favaro, Stefano & Nipoti, Bernardo, 2014. "Posterior analysis of rare variants in Gibbs-type species sampling models," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 79-98.
    10. repec:bla:biomet:v:73:y:2017:i:1:p:174-184 is not listed on IDEAS
    11. Zhu, W. & Leisen, Fabrizio, 2013. "A multivariate extension of a vector of Poisson- Dirichlet processes," DES - Working Papers. Statistics and Econometrics. WS ws132220, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. repec:bla:jorssb:v:79:y:2017:i:5:p:1295-1366 is not listed on IDEAS
    13. Ruth Fuentes–García & Ramsés Mena & Stephen Walker, 2010. "A Probability for Classification Based on the Dirichlet Process Mixture Model," Journal of Classification, Springer;The Classification Society, vol. 27(3), pages 389-403, November.
    14. repec:spr:aistmt:v:70:y:2018:i:1:d:10.1007_s10463-016-0583-8 is not listed on IDEAS

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