Bayesian Analysis

Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different

Fumiyasu Komaki

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Bayesian predictive densities when the observed data x and the target variable y to be predicted have different distributions are investigated by using the framework of information geometry. The performance of predictive densities is evaluated by the Kullback–Leibler divergence. The parametric models are formulated as Riemannian manifolds. In the conventional setting in which x and y have the same distribution, the Fisher–Rao metric and the Jeffreys prior play essential roles. In the present setting in which x and y have different distributions, a new metric, which we call the predictive metric, constructed by using the Fisher information matrices of x and y, and the volume element based on the predictive metric play the corresponding roles. It is shown that Bayesian predictive densities based on priors constructed by using non-constant positive superharmonic functions with respect to the predictive metric asymptotically dominate those based on the volume element prior of the predictive metric.

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Bayesian Anal., Volume 10, Number 1 (2015), 31-51.

First available in Project Euclid: 28 January 2015

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differential geometry Fisher–Rao metric Jeffreys prior Kullback– Leibler divergence predictive metric


Komaki, Fumiyasu. Asymptotic Properties of Bayesian Predictive Densities When the Distributions of Data and Target Variables are Different. Bayesian Anal. 10 (2015), no. 1, 31--51. doi:10.1214/14-BA886.

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  • Aitchison, J. and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge: Cambridge University Press.
  • Amari, S. (1985). Differential-Geometrical Methods in Statistics. New York: Springer-Verlag.
  • Berger, J. O. and Bernardo, J. M. (1992). “On the development of reference priors (with discussion).” In Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics 4, 35–60. New York: Oxford University Press.
  • Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge: Cambridge University Press.
  • Efron, B. (1975). “Defining curvature of a statistical problem (with applications to second order efficiency).” Annals of Statistics, 3: 1189–1242.
  • Fushiki, T., Komaki, F., and Aihara, K. (2004). “On parametric bootstrapping and Bayesian prediction.” Scandinavian Journal of Statistics, 31: 403–416.
  • Geisser, S. (1993). Predictive Inference: An Introduction. New York: Chapman & Hall.
  • George, E. I., Liang, F., and Xu, X. (2006). “Improved minimax prediction under Kullback–Leibler loss.” Annals of Statistics, 34: 78–91.
  • George, E. I. and Xu, X. (2008). “Predictive density estimation for multiple regression.” Econometric Theory, 24: 528–544.
  • Hartigan, J. A. (1998). “The maximum likelihood prior.” Annals of Statistics, 26: 2083–2103.
  • Helgason, S. (1984). Groups and Geometric Analysis. Orlando, FL: Academic Press.
  • Kobayashi, K. and Komaki, F. (2008). “Bayesian shrinkage prediction for the regression problem.” Journal of Multivariate Analysis, 99: 1888–1905.
  • Komaki, F. (1996). “On asymptotic properties of predictive distributions.” Biometrika, 83: 299–313.
  • — (2001). “A shrinkage predictive distribution for multivariate normal observables.” Biometrika, 88: 859–864.
  • — (2004). “Simultaneous prediction of independent Poisson observables.” Annals of Statistics, 32: 1744–1769.
  • — (2006). “Shrinkage priors for Bayesian prediction.” Annals of Statistics, 34: 808–819.
  • — (2007). “Bayesian prediction based on a class of shrinkage priors for location-scale models.” Annals of the Institute of Statistical Mathematics, 59: 135–146.
  • Sweeting, T. J., Datta, G. S., and Ghosh, M. (2006). “Nonsubjective priors via predictive relative entropy regret.” Annals of Statistics, 34: 441–468.
  • Zidek, J. V. (1969). “A representation of Bayesian invariant procedures in terms of Haar measure.” Annals of the Institute of Statistical Mathematics, 21: 291–308.