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Bayesian Inference via Classes of Normalized Random Measures

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Author Info

  • Lancelot F. James

    ()

  • Antonio Lijoi

    ()

  • Igor Pruenster

    ()

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    Abstract

    One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. Here we exploit theoretical properties of Poisson random measures in order to provide a comprehensive Bayesian analysis of random probabilities which are obtained by an appropriate normalization. Specifically we achieve explicit and tractable forms of the posterior and the marginal distributions, including an explicit and easily used description of generalizations of the important Blackwell-MacQueen Pólya urn distribution. Such simplifications are achieved by the use of a latent variable which admits quite interesting interpretations which allow to gain a better understanding of the behaviour of these random probability measures. It is noteworthy that these models are generalizations of models considered by Kingman (1975) in a non-Bayesian context. Such models are known to play a significant role in a variety of applications including genetics, physics, and work involving random mappings and assemblies. Hence our analysis is of utility in those contexts as well. We also show how our results may be applied to Bayesian mixture models and describe computational schemes which are generalizations of known efficient methods for the case of the Dirichlet process. We illustrate new examples of processes which can play the role of priors for Bayesian nonparametric inference and finally point out some interesting connections with the theory of generalized gamma convolutions initiated by Thorin and further developed by Bondesson.

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    File URL: http://www.icer.it/docs/wp2005/ICERwp5-05.pdf
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    Bibliographic Info

    Paper provided by ICER - International Centre for Economic Research in its series ICER Working Papers - Applied Mathematics Series with number 5-2005.

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    Length: 41 pages
    Date of creation: Apr 2005
    Date of revision:
    Handle: RePEc:icr:wpmath:5-2005

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    Related research

    Keywords: Bayesian Nonparametrics; Chinese restaurant process; Generalized gamma convolutions; Gibbs partitions; Poisson random measure;

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    References

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    1. Domenico Menicucci, 2003. "Optimal two-object auctions with synergies," Review of Economic Design, Springer, vol. 8(2), pages 143-164, October.
    2. Massimo Marinacci & Paolo Ghirardato, 2001. "Risk, ambiguity, and the separation of utility and beliefs," ICER Working Papers - Applied Mathematics Series 21-2001, ICER - International Centre for Economic Research.
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    Cited by:
    1. Cerquetti, Annalisa, 2007. "A note on Bayesian nonparametric priors derived from exponentially tilted Poisson-Kingman models," Statistics & Probability Letters, Elsevier, vol. 77(18), pages 1705-1711, December.
    2. Antonio Lijoi & Igor Pruenster & Stephen G. Walker, 2008. "Bayesian nonparametric estimators derived from conditional Gibbs structures," ICER Working Papers - Applied Mathematics Series 06-2008, ICER - International Centre for Economic Research.

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