Bayesian prediction and estimation based on a shrinkage prior for a Poisson regression model

We introduce a Poisson regression model with a natural structure, as viewed from the perspective of information geometry. A shrinkage prior distribution can be utilized similarly to the multi-dimensional Poisson distribution. In the Poisson regression model, we demonstrate that the Bayes extended estimator and Bayesian predictive density based on the shrinkage prior dominate those based on the Jeffreys prior under the Kullback-Leibler loss. By considering a multi-dimensional Poisson process, we show that the loss of the Bayesian predictive density is expressed as the integral of the loss of the Bayes extended estimators. Additionally, we discuss on the numerical evaluation of the Bayes extended estimates