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.
"Asymptotically minimax Bayesian predictive densities for multinomial models." Electron. J. Statist. 6 934 - 957, 2012. https://doi.org/10.1214/12-EJS700