Nearly minimax empirical Bayesian prediction of independent Poisson observables

TL;DR

A class of empirical Bayesian predictive distributions that dominate theBayesian predictive distribution based on the Jeffreys prior is proposed and the K-L risk is demonstrated to be less than 1.04 times the minimax lower bound.

Abstract

In this study, simultaneous predictive distributions for independent Poisson observables were considered and the performance of predictive distributions was evaluated using the Kullback-Leibler (K-L) loss. This study proposes a class of empirical Bayesian predictive distributions that dominate the Bayesian predictive distribution based on the Jeffreys prior. The K-L risk of the empirical Bayesian predictive distributions is demonstrated to be less than 1.04 times the minimax lower bound.

Figures (3)

• Figure 1: Risk difference between $p_{\mathrm{J}}(y\mid x)$ and $\hat{p}_{\alpha}(y\mid x)$ under different $\mu$ and $d$.
• Figure 2: Log values of risk difference between $p_{\mathrm{J}}(y\mid x)$ and $\hat{p}_{\alpha}(y\mid x)$, and between $p_{\mathrm{J}}(y\mid x)$ and $p_{\mathrm{S}}(y\mid x)$ under different $\mu$ for (a) $d=3$ and (b) $d=8$.
• Figure 3: Graph of $f(\lambda)$ in $[0,20]$.

Theorems & Definitions (11)

• Proposition 1
• Theorem 1
• Theorem 2
• Definition 1
• Proposition 2
• Theorem 3
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4