Annals of Statistics

Shrinkage priors for Bayesian prediction

Fumiyasu Komaki

Full-text: Open access


We investigate shrinkage priors for constructing Bayesian predictive distributions. It is shown that there exist shrinkage predictive distributions asymptotically dominating Bayesian predictive distributions based on the Jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions. Kullback–Leibler divergence from the true distribution to a predictive distribution is adopted as a loss function. Conformal transformations of model manifolds corresponding to vague priors are introduced. We show several examples where shrinkage predictive distributions dominate Bayesian predictive distributions based on vague priors.

Article information

Ann. Statist., Volume 34, Number 2 (2006), 808-819.

First available in Project Euclid: 27 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference 62C15: Admissibility

Asymptotic theory conformal transformation information geometry Jeffreys prior Kullback–Leibler divergence vague prior


Komaki, Fumiyasu. Shrinkage priors for Bayesian prediction. Ann. Statist. 34 (2006), no. 2, 808--819. doi:10.1214/009053606000000010.

Export citation


  • Aitchison, J. and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge Univ. Press.
  • Amari, S. (1985). Differential–Geometrical Methods in Statistics. Lecture Notes in Statist. 28. Springer, New York.
  • Amari, S. and Nagaoka, H. (2000). Methods of Information Geometry. Amer. Math. Soc., Providence, RI.
  • Aomoto, K. (1966). L'analyse harmonique sur les espaces riemanniens, à courbure riemannienne négative. I. J. Fac. Sci. Univ. Tokyo Sect. I 13 85–105.
  • Brandwein, A. C. and Strawderman, W. E. (1991). Generalizations of James–Stein estimators under spherical symmetry. Ann. Statist. 19 1639–1650.
  • Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855–903.
  • Brown, L. D. (1979). A heuristic method for determining admissibility of estimators–-with applications. Ann. Statist. 7 960–994.
  • Datta, G. S., Mukerjee, R., Ghosh, M. and Sweeting, T. J. (2000). Bayesian prediction with approximate frequentist validity. Ann. Statist. 28 1414–1426.
  • Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • Eguchi, S. and Yanagimoto, T. (1998). Asymptotical improvement of maximum likelihood estimators using relative entropy risk. Research Memorandum No. 665, Institute of Statistical Mathematics, Japan.
  • Eisenhart, L. P. (1949). Riemannian Geometry, 2nd ed. Princeton Univ. Press.
  • Geisser, S. (1993). Predictive Inference: An Introduction. Chapman and Hall, New York.
  • George, E. I., Liang, F. and Xu, X. (2006). Improved minimax predictive densities under Kullback–Leibler loss. Ann. Statist. 34 78–91.
  • Ghosh, J. K. (1994). Higher Order Asymptotics. IMS, Hayward, CA.
  • Hartigan, J. A. (1998). The maximum likelihood prior. Ann. Statist. 26 2083–2103.
  • Itô, S. (1964). On existence of Green function and positive superharmonic functions for linear elliptic operators of second order. J. Math. Soc. Japan 16 299–306.
  • Itô, S. (1964). Martin boundary for linear elliptic differential operators of second order in a manifold. J. Math. Soc. Japan 16 307–334.
  • Komaki, F. (1996). On asymptotic properties of predictive distributions. Biometrika 83 299–313.
  • Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observables. Biometrika 88 859–864.
  • Komaki, F. (2002). Bayesian predictive distribution with right invariant priors. Calcutta Statist. Assoc. Bull. 52 171–179.
  • Komaki, F. (2004). Simultaneous prediction of independent Poisson observables. Ann. Statist. 32 1744–1769.
  • Komaki, F. (2005). Shrinkage priors for Bayesian prediction. METR05-39, Dept. Mathematical Informatics, Univ. Tokyo.
  • Levit, B. Ya. (1982). Minimax estimation and positive solutions of elliptic equations. Theory Probab. Appl. 27 563–586.
  • Levit, B. Ya. (1983). Second-order availability and positive solutions of the Schrödinger equation. Probability Theory and Mathematical Statistics. Lecture Notes in Math. 1021 372–385. Springer, Berlin.
  • Levit, B. Ya. (1985). Second-order asymptotic optimality and positive solutions of Schrödinger's equation. Theory Probab. Appl. 30 333–363.
  • Stein, C. (1974). Estimation of the mean of a multivariate normal distribution. In Proc. Prague Symposium on Asymptotic Statistics (J. Hájek, ed.) 2 345–381. Univ. Karlova, Prague.
  • Urakawa, H. (1993). Geometry of Laplace–Beltrami operator on a complete Riemannian manifold. In Progress in Differential Geometry (K. Shiohama, ed.) 347–406. Mathematical Soc. Japan, Tokyo.
  • Vidoni, P. (1995). A simple predictive density based on the $p^*$-formula. Biometrika 82 855–863.
  • Zidek, J. V. (1969). A representation of Bayesian invariant procedures in terms of Haar measure. Ann. Inst. Statist. Math. 21 291–308.