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Objective Bayesian analysis under sequential experimentation

Dongchu Sun and James O. Berger

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Abstract

Objective priors for sequential experiments are considered. Common priors, such as the Jeffreys prior and the reference prior, will typically depend on the stopping rule used for the sequential experiment. New expressions for reference priors are obtained in various contexts, and computational issues involving such priors are considered.

Chapter information

Source
Bertrand Clarke and Subhashis Ghosal, eds., Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 19-32

Dates
First available in Project Euclid: 28 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1209398457

Digital Object Identifier
doi:10.1214/074921708000000020

Subjects
Primary: 62L12: Sequential estimation 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62F15: Bayesian inference 62L10: Sequential analysis

Keywords
expected stopping time frequentist coverage Jeffreys’ prior posterior distributions reference prior sequential experimentation

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

Sun, Dongchu; Berger, James O. Objective Bayesian analysis under sequential experimentation. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 19--32, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000020. https://projecteuclid.org/euclid.imsc/1209398457


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References

  • [1] Bar-Lev, S. K. and Reiser, B. (1982). An exponential subfamily which admits UMPU test based on a single test statistic. Ann. Statist. 10 979–989.
  • [2] Bartholomew, D. (1965). A comparison of some Bayesian and frequentist inference. Biometrika 52 19–35.
  • [3] Berger, J. O. (2006). The case for objective Bayesian analysis. Bayesian Analysis 1 385–402 and 457–464.
  • [4] Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors. J. Amer. Statist. Assoc. 84 200–207.
  • [5] Berger, J. O. and Bernardo, J. M. (1992). On the development of the reference prior method (with discussion). In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 35–60. Oxford Univ. Press.
  • [6] Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 41 113–147.
  • [7] Bernardo, J. M. (2005). Reference analysis. In Handbook of Statistics 25 (D. K. Dey and C. R. Rao, eds.) 17–90. North-Holland, Amsterdam.
  • [8] Bernardo, J. M. and Smith, A. F. M. (1984). Bayesian Theory. Wiley, New York.
  • [9] Bose, A. and Boukai, B. (1993). Sequential estimation results for a two-parameter exponential family of distributions. Ann. Statist. 21 484–502.
  • [10] Brown, L. D. (1988), The differential inequality of a statistical estimation problem, Statistical Decision Theory and Related Topics IV 1 299–324.
  • [11] Cox, D. R. and Reid, N. (1987). Orthogonal parameters and approximate conditional inference (with discussion). J. Roy. Statist. Soc. Ser. B 49 1–39.
  • [12] Datta, G. S. (1996). On priors providing frequentist validity for Bayesian inference for multiple parametric functions. Biometrika 83 287–298.
  • [13] Datta, G. S. and Ghosh, J. K. (1995). On priors providing frequentist validity for Bayesian inference. Biometrika 82 37–45.
  • [14] Datta, G. S. and Ghosh, M. (1995). Some remarks on noninformative priors J. Amer. Statist. Assoc. 90 1357–1363.
  • [15] Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. Ann. Statist. 24 141–159.
  • [16] Datta, G. S., Ghosh, M. and Mukerjee, R. (2000). Some new results on probability matching priors. Calcutta Statist. Assoc. Bull. 50 179–192.
  • [17] Datta, G. S. and Mukerjee, R. (2004). Probability Matching Priors: Higher Order Asymptotics. Springer, New York.
  • [18] Geisser, S. (1979). Comments on “Reference posterior distributions for Bayesian inference”, by J. Bernardo. J. Roy. Statist. Soc. Ser. B 41 136–137.
  • [19] Geisser, S. (1984). On prior distributions for binary trials. American Statisticians 38 244–251.
  • [20] Ghosh, J. K. (1994). Higher Order Asymptotics. IMS and Amer. Statist. Assoc., Hayward, CA.
  • [21] Ghosh, J. K., Delampady, M. and Samanta, T. (2006). An Introduction to Bayesian Analysis: Theory and Methods. Springer, New York.
  • [22] Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P Dawid and A. F. M. Smith, eds.) 195–210. Oxford Univ. Press.
  • [23] Ghosh, J. K. and Mukerjee, R. (1995). Frequentist validity of highest posterior density regions in the presence of nuisance parameters. Statist. Dec. 13 131–139.
  • [24] Ghosh, M., Sen, P. K. and Mukhopadhyay, N. (1997). Sequential Estimation. Wiley, New York.
  • [25] Govindarajulu, Z. (1981). The Sequential Statistical Analysis of Hypothesis Testing, Point and Interval Estimation, and Decision Theory. American Science Press, Columbus, OH.
  • [26] Hall, W. J. (1992). A course in sequential analysis. Unpublished Lecture Notes, University of Rochester, Rochester, NY.
  • [27] Jeffreys, H. (1961). Theory of Probability. Oxford Univ. Press.
  • [28] Peers, H. W. (1965). On confidence sets and Bayesian probability points in the case of several parameters. J. Roy. Statist. Soc. Ser. B 27 9–16.
  • [29] Polson, N. and Roberts, G. (1993). A utility based approach to information for stochastic differential equations. Stochastic Proc. Appl. 48 341–356.
  • [30] Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals. Springer, New York.
  • [31] Sivaganesan, S. and Lingam, R. (2002). Bayes Factors for model selection with diffusion processes under improper priors. Ann. Instit. Statist. Math. 54 500–516.
  • [32] Sun, D. (1994). Integrable expansions for posterior distributions for a two-parameter exponential family. Ann. Statist. 22 1808–1830.
  • [33] Sun, D. and Ye, K. (1996). Frequentist validity of posterior quantiles for a two-parameter exponential family. Biometrika 83 55–65.
  • [34] Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika 76 604–608.
  • [35] Walker, A. M. (1969). On the asymptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80–88.
  • [36] Welch, B. N. and Peers, B. (1963). On formulae for confidence points based on integrals of weighted likelihoods. J. Roy. Statist. Soc. Ser. B 35 318–329.
  • [37] Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia.
  • [38] Ye, K. (1993). Reference priors when the stopping rule depends on the parameter of interest. J. Amer. Statist. Assoc. 88 360–363.