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Asymptotic Normality with Small Relative Errors of Posterior Probabilities of Half-Spaces

  • R. M. Dudley
  • D. Haughton
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

Some examples of half-spaces of interest in parameter spaces are, in a clinical trial of a treatment versus placebo, the half-spaces where the treatment is (a) helpful or (b) harmful. Thus one may not only want to test the hypothesis that the treatment (c) makes no difference, but to assign posterior probabilities to (a), (b) and (c), under conditions as unrestrictive as possible on the choice of prior probabilities [e.g., Dudley and Haughton (2001)]. More generally, we have in mind applications to model selections as in the BIC criterion of Schwarz (1978) and its extensions [Poskitt (1987); Haughton (1988)], specifically to one-sided models and multiple data sets [Dudley and Haughton (1997)].

Key words and phrases

Bernstein–von Mises theorem gamma tail probabilities intermediate deviations Jeffreys prior Mills’ ratio 

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References

  1. Alon, N., Ben-David, S., Cesa-Bianchi, N. and Haussler, D. (1997). Scale-sensitive dimensions, uniform convergence, and learnability. J. ACM 44 615–631.MATHCrossRefMathSciNetGoogle Scholar
  2. Barndorff-Nielsen, O. E. and Wood, A. T. A. (1998). On large deviations and choice of ancillary for p * and r *. Bernoulli 4 35–63.MATHCrossRefMathSciNetGoogle Scholar
  3. Berger, J. O. and Mortera, J. (1999). Default Bayes factors for nonnested hypothesis testing. J. Amer. Statist. Assoc. 94 542–554.MATHCrossRefMathSciNetGoogle Scholar
  4. Berk, R. H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist. 37 51–58. [Correction (1996) 37 745–746.]MATHCrossRefMathSciNetGoogle Scholar
  5. Bickel, P. J. and Ghosh, J. K. (1990). A decomposition for the likelihood ratio statistic and the Bartlett correction—a Bayesian argument. Ann. Statist. 18 1070–1090.MATHCrossRefMathSciNetGoogle Scholar
  6. Bleistein, N. (1966). Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19 353–370.MATHCrossRefMathSciNetGoogle Scholar
  7. Chen, C.-F. (1985). On asymptotic normality of limiting density functions with Bayesian implications. J. Roy. Statist. Soc. Ser. B 47 540–546.MATHMathSciNetGoogle Scholar
  8. Choy, S. T. B. and Smith, A. F. M. (1997). On robust analysis of a normal location parameter. J. Roy. Statist. Soc. Ser. B 59 463–474.MATHCrossRefMathSciNetGoogle Scholar
  9. Daniels, H. E. (1987). Tail probability approximations. Internat. Statist. Rev. 55 37–48.MATHCrossRefMathSciNetGoogle Scholar
  10. DiCiccio, T. J. and Martin, M. A. (1991). Approximations of marginal tail probabilities for a class of smooth functions with applications to Bayesian and conditional inference. Biometrika 78 891–902.MATHCrossRefMathSciNetGoogle Scholar
  11. DiCiccio, T. J. and Stern, S. E. (1993). On Bartlett adjustments for approximate Bayesian inference. Biometrika 80 731–740.MATHCrossRefMathSciNetGoogle Scholar
  12. Dudley, R. M. (1993). Real Analysis and Probability, 2nd ed., corrected. Chapman and Hall, New York.Google Scholar
  13. Dudley, R. M. (1998). Consistency of M-estimators and one-sided bracketing. In High Dimensional Probability (E. Eberlein, M. Hahn and M. Talagrand, eds.) 33–58. Birkhäuser, Basel.Google Scholar
  14. Dudley, R. M., Giné, E. and Zinn, J. (1991). Uniform and universal Glivenko–Cantelli classes. J. Theoret. Probab. 4 485–510.MATHCrossRefMathSciNetGoogle Scholar
  15. Dudley, R. M. and Haughton, D. (1997). Information criteria for multiple data sets and restricted parameters. Statist. Sinica 7 265–284.MATHMathSciNetGoogle Scholar
  16. Dudley, R. M. and Haughton, D. (2001). One-sided hypotheses in a multinomial model. In Goodness-of-Fit Tests and Model Validity (C. Huber-Carol, N. Balakrishnan, M. S. Nikulin and M. Mesbah, eds.) 387–399. Birkhäuser, Boston.Google Scholar
  17. Erkanli, A. (1994). Laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space. J. Amer. Statist. Assoc. 89 250–258.MATHCrossRefMathSciNetGoogle Scholar
  18. Fraser, D. A. S., Reid, N. and Wu, J. (1999). A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86 249–264.MATHCrossRefMathSciNetGoogle Scholar
  19. Fulks, W. and Sather, J. O. (1961). Asymptotics II Laplace’s method for multiple integrals. Pacific J. Math. 11 185–192.MATHMathSciNetGoogle Scholar
  20. Haughton, D. (1984). On the choice of a model to fit data from an exponential family. Ph.D. dissertation, MIT.Google Scholar
  21. Haughton, D. M. A. (1988). On the choice of a model to fit data from an exponential family. Ann. Statist. 16 342–355.MATHCrossRefMathSciNetGoogle Scholar
  22. Hipp C. and Michel, R. (1976). On the Bernstein–v. Mises approximation of posterior distributions. Ann. Statist. 4 972–980.MATHCrossRefMathSciNetGoogle Scholar
  23. Holt, R. J. (1986). Computation of gamma and beta tail probabilities. Technical report, Dept. Mathematics, MIT.Google Scholar
  24. Hsu, L. C. (1948). A theorem on the asymptotic behavior of a multiple integral. Duke Math. J. 15 623–632.MATHCrossRefMathSciNetGoogle Scholar
  25. Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 221–233. Univ. California Press, Berkeley.Google Scholar
  26. Jensen, J. L. (1995). Saddlepoint Approximations. Oxford Univ. Press.Google Scholar
  27. Johnson, R. A. (1970). Asymptotic expansions associated with posterior distributions. Ann. Math. Statist. 41 851–864.MATHCrossRefMathSciNetGoogle Scholar
  28. Lawley, D. N. (1956). A general method for approximating to the distribution of likelihood ratio criteria. Biometrika 43 295–303.MATHMathSciNetGoogle Scholar
  29. Le Cam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Bayes’ estimates. Univ. California Publ. Statist. 1 227–329.MathSciNetGoogle Scholar
  30. Leonard, T. (1982). Comment. J. Amer. Statist. Assoc., 77 657–658.CrossRefMathSciNetGoogle Scholar
  31. Lugannani, R. and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Adv. in Appl. Probab. 12 475–490.MATHCrossRefMathSciNetGoogle Scholar
  32. Pauler, D. K., Wakefield, J. C. and Kass, R. E. (1999). Bayes factors and approximations for variance component models. J. Amer. Statist. Assoc. 94 1242–1253.MATHCrossRefMathSciNetGoogle Scholar
  33. Pericchi, L. R. and Smith, A. F. M. (1992). Exact and approximate posterior moments for a normal location parameter. J. Roy. Statist. Soc. Ser. B. 54 793–804.MATHMathSciNetGoogle Scholar
  34. Poskitt, D. S. (1987). Precision, complexity and Bayesian model determination. J. Roy. Statis.t Soc. Ser. B. 49 199–208.MATHMathSciNetGoogle Scholar
  35. Reid, N. (1996). Likelihood and higher-order approximations to tail areas: A review and annotated bibliography. Canad. J. Statist. 24 141–166.MATHCrossRefMathSciNetGoogle Scholar
  36. Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, New York.MATHGoogle Scholar
  37. Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist 6 461–464.MATHCrossRefMathSciNetGoogle Scholar
  38. Shun, Z. and McCullagh, P. (1995). Laplace approximation of high-dimensional integrals. J. Roy. Statist. Soc. Ser. B 57 749–760.MATHMathSciNetGoogle Scholar
  39. Skinner, L. A. (1980). Note on the asymptotic behavior of multidimensional Laplace integrals. SIAM J. Math. Anal. 11 911–917.MATHCrossRefMathSciNetGoogle Scholar
  40. Talagrand, M. (1987). The Glivenko–Cantelli problem. Ann. Probab. 15 837–870.MATHCrossRefMathSciNetGoogle Scholar
  41. Temme, N. M. (1982). The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions. SIAM J. Math. Anal. 13 239–253.MATHCrossRefMathSciNetGoogle Scholar
  42. Temme, N. M. (1987). Incomplete Laplace integrals: Uniform asymptotic expansion with application to the incomplete beta function. SIAM J. Math. Anal. 18 1638–1663.MATHCrossRefMathSciNetGoogle Scholar
  43. Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. J. Amer. Statist. Assoc. 81 82–86.MATHCrossRefMathSciNetGoogle Scholar
  44. Tierney, L., Kass, R. E. and Kadane, J. B. (1989). Approximate marginal densities of nonlinear functions. Biometrika 76 425–433. [Correction (1991) 78 233–234.]MATHCrossRefMathSciNetGoogle Scholar
  45. van der Vaart, A. W. and Wellner, J. A. (2000). Preservation theorems for Glivenko–Cantelli and uniform Glivenko–Cantelli classes. In High Dimensional Probability II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 115–133. Birkhäuser, Boston.Google Scholar
  46. Walker, A. M. (1969). On the asymptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80–88.MATHMathSciNetGoogle Scholar
  47. Wall, H. S. (1948). Analytic Theory of Continued Fractions. Van Nostrand, New York.MATHGoogle Scholar
  48. Wong, R. (1973). On uniform asymptotic expansion of definite integrals. J. Approx. Theory 7 76–86.MATHCrossRefGoogle Scholar
  49. Wong, R. (1989). Asymptotic Approximations of Integrals. Academic Press, New York.MATHGoogle Scholar
  50. Woodroofe, M. (1992). Integrable expansions for posterior distributions for one-parameter exponential families. Statist. Sinica 2 91–111.MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Mathematical SciencesBentley CollegeWalthamUSA

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