Asymptotically minimax Bayesian predictive densities for multinomial models

TL;DR

This paper investigates Bayesian predictive densities for multinomial models, deriving asymptotic risk approximations and identifying a prior that achieves asymptotic minimaxity, distinct from classical objective priors.

Contribution

It introduces a new Dirichlet prior that yields an asymptotically minimax Bayesian predictive density for multinomial models.

Findings

Derived asymptotic risk functions for Bayesian predictive densities.

Identified a specific Dirichlet prior that is asymptotically minimax.

Showed this prior differs from known objective priors like Jeffreys or uniform.

Abstract

One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.