  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.
