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A Generalized Bayes Rule for Prediction

Author

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  • José Manuel Corcuera
  • Federica Giummolè

Abstract

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback‐Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α‐divergence, including the Kullback‐Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non‐Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.

Suggested Citation

  • José Manuel Corcuera & Federica Giummolè, 1999. "A Generalized Bayes Rule for Prediction," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 265-279, June.
  • Handle: RePEc:bla:scjsta:v:26:y:1999:i:2:p:265-279
    DOI: 10.1111/1467-9469.00149
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    Citations

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    Cited by:

    1. Chang, In Hong & Mukerjee, Rahul, 2004. "Asymptotic results on the frequentist mean squared error of generalized Bayes point predictors," Statistics & Probability Letters, Elsevier, vol. 67(1), pages 65-71, March.
    2. Malay Ghosh & Tatsuya Kubokawa & Gauri Sankar Datta, 2020. "Density Prediction and the Stein Phenomenon," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 330-352, August.
    3. Takemi Yanagimoto & Toshio Ohnishi, 2014. "Permissible boundary prior function as a virtually proper prior density," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 789-809, August.
    4. Ghosh, Malay & Mergel, Victor & Datta, Gauri Sankar, 2008. "Estimation, prediction and the Stein phenomenon under divergence loss," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1941-1961, October.
    5. Abdolnasser Sadeghkhani, 2022. "On Improving the Posterior Predictive Distribution of the Difference Between two Independent Poisson Distribution," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 765-777, November.
    6. Tatsuya Kubokawa & Éric Marchand & William E. Strawderman & Jean-Philippe Turcotte, 2012. "Minimaxity in Predictive Density Estimation with Parametric Constraints," CIRJE F-Series CIRJE-F-843, CIRJE, Faculty of Economics, University of Tokyo.
    7. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E. & Turcotte, Jean-Philippe, 2013. "Minimaxity in predictive density estimation with parametric constraints," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 382-397.
    8. Zhang, Fode & Shi, Yimin & Wang, Ruibing, 2017. "Geometry of the q-exponential distribution with dependent competing risks and accelerated life testing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 552-565.
    9. Essam Al-Hussaini & Abd Ahmad, 2003. "On Bayesian interval prediction of future records," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(1), pages 79-99, June.

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