Electronic Journal of Statistics

Asymptotically minimax Bayesian predictive densities for multinomial models

Fumiyasu Komaki

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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.

Article information

Electron. J. Statist., Volume 6 (2012), 934-957.

First available in Project Euclid: 25 May 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures 62C20: Minimax procedures
Secondary: 62F15: Bayesian inference

Dirichlet prior Jeffreys prior Kullback-Leibler divergence latent information prior reference prior


Komaki, Fumiyasu. Asymptotically minimax Bayesian predictive densities for multinomial models. Electron. J. Statist. 6 (2012), 934--957. doi:10.1214/12-EJS700. https://projecteuclid.org/euclid.ejs/1337951629

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  • [1] Aitchison, J. (1975). Goodness of prediction fit., Biometrika, 62, 547–554.
  • [2] Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors., Journal of the American Statistical Association, 84, 200–207.
  • [3] Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion)., Journal of Royal Statistical Society B, 41, 113–147.
  • [4] Bernardo, J. M. (2005). Reference analysis., Handbook of Statistics, 25, Dey, K. K. and Rao C. R. eds., Elsevier, Amsterdam 17–90.
  • [5] Clarke, B. (2007). Information optimality and Bayesian modelling., Journal of Econometrics, 138, 405–429.
  • [6] Clarke, B. and Barron, A. R. (1994). Jeffreys’ prior is asymptotically least favorable under entropy risk., Journal of Statistical Planning and Inference, 41, 36–60.
  • [7] Grünwald, P. D. and Dawid, A. P. (2004). Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory., Annals of Statistics, 32, 1367–1433.
  • [8] Ibragimov, I. A. and Hasminskii, R. Z. (1973). On the information contained in a sample about a parameter. In, 2nd Intl. Symp. on Information Theory, Akademiai, Kiado, Budapest 295–309.
  • [9] Komaki, F. (1996). On asymptotic properties of predictive distributions., Biometrika, 83, 299–313.
  • [10] Komaki, F. (2004). Simultaneous prediction of independent Poisson observables., Annals of Statistics, 32, 1744–1769.
  • [11] Komaki, F. (2011). Bayesian predictive densities based on latent information priors., Journal of Statistical Planning and Inference, 141, 3705–3715.
  • [12] Romanovsky, V. (1923). Note on the moments of a binomial, (p+q)n about its mean, Biometrika, 15, 410–412.
  • [13] Xie, Q. and Barron, A. R. (1997). Minimax redundancy for the class of memoryless sources., IEEE Transactions on Information Theory, 43, 646–657.
  • [14] Xie, Q. and Barron, A. R. (2000). Asymptotic minimax regret for data compression, gambling, and prediction., IEEE Transaction on Information Theory, 46, 431–445.