Approximate Bayesian methods

D. V. Lindley1
1University College London, London, UK

Tóm tắt

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.

Tài liệu tham khảo

Anderson, T.W. (1958).An introduction to multivariate statistical analysis. New York: Wiley Barndorff-Nielsen, O. andCox, D.R. (1979). Edgeworth and saddlepoint approximations with statistical applications.J. Roy. Statist. Soc B,41, 279–312 Deely, J.J. andLindley, D.V. (1979). Bayes empirical Bayes.Tech. Report. University of Canterbury. Dunsmore, I.R. (1976). Asymptotic prediction analysis.Biometrika,63, 627–630. Jeffreys, H. (1961)Theory of probability. Oxford: Clarendon Press. Lindley, D.V. (1961). The use of prior probability distributions in statistical inference and decisions.Proc 4th Berkeley Symp.1, 453–468. Blum, E.K. (1972)Numerical Analysis and Computation, Reading, Mass. Addison-Wesley Dawid, A.P. (1973) Posterior expectations for large observations.Biometrika,60, 664–666. Efron, B. andMorris, C. (1973) Stein’s estimation rule and its competitors-an empirical Bayes approach.J. Amer. Statist. Assoc. 68, 117–30. Efron, B. (1975) Defining the curvature of a statistical problem (with applications to second order efficiency).Ann. Statist. 3, 1189–1217. Fox, L. andMayers, D.F. (1968)Computing Methods for Scientists and Engineers. Oxford: University Press. Fröberg, C.E. (1969)Introduction to Numerical Analysis (2nd edn) Reading, Mass: Addison-Wesley Goldstein, M. (1975a). Approximate Bayes solutions to some non-parametric problems.Ann. Statist. 3, 512–517. — (1975b). A note on some Bayesian non-parametric problems.Ann. Statist. 3, 736–740. — (1976). Bayesian analysis of regression problems.Biometrika 63, 51–58. Good, I.J. andGaskins, R.A. (1969) The centroid method of integration.Nature 222, 697–698 — (1971) The centroid method of numerical integration.Numerische Mathematik 16, 343–359. Good, I.J. andTideman, T.N. (1978) Integration over a simplex, truncated cubes, and Eulerian numbers.Numerische Mathematik,30, 355–367 Hartigan, J.A. (1969). Linear Bayesian methods.J. Roy. Statist. Soc. B,31, 446–454. 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. Kloek, T. andVan Dijk, H.K. (1978). Bayesian estimates of equation system parameters. An application of Integration by Monte-Carlo.Econometrica,46, 1–19. Lindley, D.V. (1975) Comments on Efron (1975)Ann. Statist. 3, 1222–1223.