Information Geometry of Generalized Bayesian Prediction Using $\alpha$ -Divergences as Loss Functions

Abstract

In this paper, the methods of information geometry are employed to investigate a generalized Bayes rule for prediction. Taking α-divergences as the loss functions, optimality, and asymptotic properties of the generalized Bayesian predictive densities are considered. We show that the Bayesian predictive densities minimize a generalized Bayes risk. We also find that the asymptotic expansions of the densities are related to the coefficients of the α-connections of a statistical manifold. In addition, we discuss the difference between two risk functions of the generalized Bayesian predictions based on different priors. Finally, using the non-informative priors (i.e., Jeffreys and reference priors), uniform prior, and conjugate prior, two examples are presented to illustrate the main results.