A Generalized Bayes Rule for Prediction

A Generalized Bayes Rule for Prediction 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Scandinavian Journal of Statistics Wiley

A Generalized Bayes Rule for Prediction

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Publisher
Wiley
Copyright
Board of the Foundation of the Scandinavian Journal of Statistics 1999
ISSN
0303-6898
eISSN
1467-9469
D.O.I.
10.1111/1467-9469.00149
Publisher site
See Article on Publisher Site

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.

Journal

Scandinavian Journal of StatisticsWiley

Published: Jun 1, 1999

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