Extended Plugin Densities for Curved Exponential Families
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Extended plugin densities are proposed as predictive densities for curved exponential families. Bayesian predictive densities are often intractable in numerical calculations, although Bayesian predictive densities are optimal under the Bayes risk based on the Kullback-Leibler divergence. Curved exponential families are embedded in larger full exponential families and plugin densities in the full exponential families, which we call extended plugin densities, are considered. It is shown that the extended plugin density with the posterior mean of the expectation parameter of the full exponential family is optimal regarding the Bayes risk in the extended model. Several information-geometric results for extended plugin densities are obtained in parallel with those for Bayesian predictive densities. Choice of priors for extended plugin densities with Bayes estimators is investigated and a superharmonic condition for a prior to dominate the Jeffreys prior is obtained.
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