Improved nearly minimax prediction for independent Poisson processes under Kullback-Leibler loss

TL;DR

This paper develops improved Bayesian predictive distributions for independent Poisson variables under Kullback-Leibler loss, achieving near-minimax performance by leveraging superharmonic priors and extending to Poisson processes with different durations.

Contribution

It introduces sufficient conditions based on superharmonicity to enhance Bayesian predictions for Poisson models, including cases with different process durations, with theoretical risk bounds.

Findings

Improved predictive distributions have K-L risk less than 1.04 times the minimax lower bound.

Superharmonic priors enhance Bayesian prediction performance.

Examples include point and subspace shrinkage priors.

Abstract

The problem of predicting independent Poisson random variables is commonly encountered in real-life practice. Simultaneous predictive distributions for independent Poisson observables are investigated, and the performance of predictive distributions is evaluated using the Kullback-Leibler (K-L) loss. This study introduces intuitive sufficient conditions, based on superharmonicity of priors, to improve the Bayesian predictive distribution based on the Jeffreys prior. The sufficient conditions exhibit a certain analogy with those known for the multivariate normal distribution. Additionally, this study examines the case where the observed data and target variables to be predicted are independent Poisson processes with different durations. Examples that satisfy the sufficient conditions are provided, including point and subspace shrinkage priors. The K-L risk of the improved predictions is demonstrated to be less than 1.04 times a minimax lower bound.