Bayesian predictive distribution for a Poisson model with a parametric restriction

Abstract

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.

Keywords:

• Bayes
• dominance
• estimation
• Kullback-Leibler divergence
• prediction
• restricted parameter space

Acknowledgments

We would like to thank the Editor and the three reviewers for many valuable comments and helpful suggestions which led to an improved version of this paper. Research of the second author was supported in part by Grant-in-Aid for Scientific Research (18K11188, 15H01943 and 26330036) from Japan Society for the Promotion of Science.