Bayesian prediction of a density function in terms of e-mixture

The optimum Bayesian predictor under the e-divergence loss is proposed and discussed. Notable dualistic structure is observed between the proposed predictor and the optimum predictor under the m-divergence loss, the latter of which is dominantly discussed in the existing literature. An advantage of the proposed optimum predictor is that it is estimative, when the sampling density is in the exponential family. Potential advantages of the proposed predictor over its dual one are discussed, which include the shrinkage estimator and the Bayesian model selection criterion DIC (deviance information criterion). Further, we emphasize potential usefulness of the use of Jeffreys' prior.