Advertisement

Approximate Bayesian methods

  • D. V. Lindley
Approximations Invited Papers

Summary

This paper develops asymptotic expansions for the ratios of integrals that occur in Bayesian analysis: for example, the posterior mean. The first term omitted isO(n −2) and it is shown how the termO(n −1) can be of importance.

Keywords

Asymptotic Expansions Steepest Descent Bayesian Methods Posterior Moments One-Way Analysis of Variance 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T.W. (1958).An introduction to multivariate statistical analysis. New York: WileyzbMATHGoogle Scholar
  2. Barndorff-Nielsen, O. andCox, D.R. (1979). Edgeworth and saddlepoint approximations with statistical applications.J. Roy. Statist. Soc B,41, 279–312zbMATHMathSciNetGoogle Scholar
  3. Deely, J.J. andLindley, D.V. (1979). Bayes empirical Bayes.Tech. Report. University of Canterbury.Google Scholar
  4. Dunsmore, I.R. (1976). Asymptotic prediction analysis.Biometrika,63, 627–630.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Jeffreys, H. (1961)Theory of probability. Oxford: Clarendon Press.zbMATHGoogle Scholar
  6. Lindley, D.V. (1961). The use of prior probability distributions in statistical inference and decisions.Proc 4th Berkeley Symp.1, 453–468.MathSciNetGoogle Scholar

References in the Discussion

  1. Blum, E.K. (1972)Numerical Analysis and Computation, Reading, Mass. Addison-WesleyzbMATHGoogle Scholar
  2. Dawid, A.P. (1973) Posterior expectations for large observations.Biometrika,60, 664–666.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Efron, B. andMorris, C. (1973) Stein’s estimation rule and its competitors-an empirical Bayes approach.J. Amer. Statist. Assoc. 68, 117–30.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Efron, B. (1975) Defining the curvature of a statistical problem (with applications to second order efficiency).Ann. Statist. 3, 1189–1217.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Fox, L. andMayers, D.F. (1968)Computing Methods for Scientists and Engineers. Oxford: University Press.zbMATHGoogle Scholar
  6. Fröberg, C.E. (1969)Introduction to Numerical Analysis (2nd edn) Reading, Mass: Addison-WesleyGoogle Scholar
  7. Goldstein, M. (1975a). Approximate Bayes solutions to some non-parametric problems.Ann. Statist. 3, 512–517.zbMATHCrossRefMathSciNetGoogle Scholar
  8. — (1975b). A note on some Bayesian non-parametric problems.Ann. Statist. 3, 736–740.zbMATHCrossRefMathSciNetGoogle Scholar
  9. — (1976). Bayesian analysis of regression problems.Biometrika 63, 51–58.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Good, I.J. andGaskins, R.A. (1969) The centroid method of integration.Nature 222, 697–698CrossRefGoogle Scholar
  11. — (1971) The centroid method of numerical integration.Numerische Mathematik 16, 343–359.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Good, I.J. andTideman, T.N. (1978) Integration over a simplex, truncated cubes, and Eulerian numbers.Numerische Mathematik,30, 355–367zbMATHCrossRefMathSciNetGoogle Scholar
  13. Hartigan, J.A. (1969). Linear Bayesian methods.J. Roy. Statist. Soc. B,31, 446–454.zbMATHMathSciNetGoogle Scholar
  14. Hill, B.M. (1974) On coherence inadmissibility and inference about many parameters in the theory of least squares. InStudies in Bayesian Econometrics and Statistics, (Fienberg, S.E. and Zellner, A. eds.) Amsterdam: North-Holland.Google Scholar
  15. Kloek, T. andVan Dijk, H.K. (1978). Bayesian estimates of equation system parameters. An application of Integration by Monte-Carlo.Econometrica,46, 1–19.zbMATHCrossRefGoogle Scholar
  16. Lindley, D.V. (1975) Comments on Efron (1975)Ann. Statist. 3, 1222–1223.MathSciNetGoogle Scholar

Copyright information

© Springer 1980

Authors and Affiliations

  • D. V. Lindley
    • 1
  1. 1.University College LondonLondonUK

Personalised recommendations