Statistical Science

The Interplay of Bayesian and Frequentist Analysis

M. J. Bayarri and J. O. Berger

Full-text: Open access


Statistics has struggled for nearly a century over the issue of whether the Bayesian or frequentist paradigm is superior. This debate is far from over and, indeed, should continue, since there are fundamental philosophical and pedagogical issues at stake. At the methodological level, however, the debate has become considerably muted, with the recognition that each approach has a great deal to contribute to statistical practice and each is actually essential for full development of the other approach. In this article, we embark upon a rather idiosyncratic walk through some of these issues.

Article information

Statist. Sci. Volume 19, Number 1 (2004), 58-80.

First available in Project Euclid: 14 July 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Admissibility Bayesian model checking conditional frequentist confidence intervals consistency coverage design hierarchical models nonparametric Bayes objective Bayesian methods p-values reference priors testing


Bayarri, M. J.; Berger, J. O. The Interplay of Bayesian and Frequentist Analysis. Statist. Sci. 19 (2004), no. 1, 58--80. doi:10.1214/088342304000000116.

Export citation


  • Barnett, V. (1982). Comparative Statistical Inference, 2nd ed. Wiley, New York.
  • Barron, A. (1999). Information-theoretic characterization of Bayes performance and the choice of priors in parametric and nonparametric problems. In Bayesian Statistics 6 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 27--52. Oxford Univ. Press.
  • Barron, A., Schervish, M. J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536--561.
  • Bayarri, M. J. and Berger, J. (2000). P-values for composite null models (with discussion). J. Amer. Statist. Assoc. 95 1127--1170.
  • Bayarri, M. J. and Castellanos, M. E. (2004). Bayesian checking of hierarchical models. Technical report, Univ. Valencia.
  • Belitser, E. and Ghosal, S. (2003). Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution. Ann. Statist. 31 536--559.
  • Berger, J. (1985a), Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
  • Berger, J. (1985b). The frequentist viewpoint and conditioning. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. Le Cam and R. Olshen, eds.) 15--44. Wadsworth, Monterey, CA.
  • Berger, J. (1994). An overview of robust Bayesian analysis (with discussion). Test 3 5--124.
  • Berger, J. (2003). Could Fisher, Jeffreys and Neyman have agreed on testing (with discussion)? Statist. Sci. 18 1--32.
  • Berger, J. and Bernardo, J. (1992). On the development of reference priors. In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 35--60. Oxford Univ. Press.
  • Berger, J. and Berry, D. (1988). The relevance of stopping rules in statistical inference (with discussion). In Statistical Decision Theory and Related Topics IV (S. Gupta and J. Berger, eds.) 1 29--72. Springer, New York.
  • Berger, J., Boukai, B. and Wang, Y. (1999). Simultaneous Bayesian--frequentist sequential testing of nested hypotheses. Biometrika 86 79--92.
  • Berger, J., Brown, L. D. and Wolpert, R. (1994). A unified conditional frequentist and Bayesian test for fixed and sequential simple hypothesis testing. Ann. Statist. 22 1787--1807.
  • Berger, J., Ghosh, J. K. and Mukhopadhyay, N. (2003). Approximations and consistency of Bayes factors as model dimension grows. J. Statist. Plann. Inference 112 241--258.
  • Berger, J. and Pericchi, L. (2001). Objective Bayesian methods for model selection: Introduction and comparison (with discussion). In Model Selection (P. Lahiri, ed.) 135--207. IMS, Beachwood, OH.
  • Berger, J. and Pericchi, L. (2004). Training samples in objective Bayesian model selection. Ann. Statist. 32 841--869.
  • Berger, J., Philippe, A. and Robert, C. (1998). Estimation of quadratic functions: Noninformative priors for non-centrality parameters. Statist. Sinica 8 359--376.
  • Berger, J. and Robert, C. (1990). Subjective hierarchical Bayes estimation of a multivariate normal mean: On the frequentist interface. Ann. Statist. 18 617--651.
  • Berger, J. and Strawderman, W. (1996). Choice of hierarchical priors: Admissibility in estimation of normal means. Ann. Statist. 24 931--951.
  • Berger, J. and Wolpert, R. L. (1988). The Likelihood Principle: A Review, Generalizations, and Statistical Implications, 2nd ed. IMS, Hayward, CA. (With discussion.)
  • Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 41 113--147.
  • Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. Wiley, New York.
  • Box, G. E. P. (1980). Sampling and Bayes' inference in scientific modeling and robustness (with discussion). J. Roy. Statist. Soc. Ser. A 143 383--430.
  • Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855--903.
  • Brown, L. D. (1994). Minimaxity, more or less. In Statistical Decision Theory and Related Topics V (S. Gupta and J. Berger, eds.) 1--18. Springer, New York.
  • Brown, L. D. (2000). An essay on statistical decision theory. J. Amer. Statist. Assoc. 95 1277--1281.
  • Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion (with discussion). Statist. Sci. 16 101--133.
  • Brown, L. D., Cai, T. T. and DasGupta, A. (2002). Confidence intervals for a binomial proportion and asymptotic expansions. Ann. Statist. 30 160--201.
  • Carlin, B. P. and Louis, T. A. (2000). Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Chapman and Hall, London.
  • Casella, G. (1988). Conditionally acceptable frequentist solutions (with discussion). In Statistical Decision Theory and Related Topics IV (S. Gupta and J. Berger, eds.) 1 73--117. Springer, New York.
  • Castellanos, M. E. (2002). Diagnóstico Bayesiano de modelos. Ph.D. dissertation, Univ. Miguel Hernández, Spain.
  • Chaloner, K. and Verdinelli, I. (1995). Bayesian experimental design: A review. Statist. Sci. 10 273--304.
  • Clyde, M. and George, E. (2004). Model uncertainty. Statist. Sci. 19 81--94.
  • Daniels, M. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89 553--566.
  • Dass, S. and Berger, J. (2003). Unified conditional frequentist and Bayesian testing of composite hypotheses. Scand. J. Statist. 30 193--210.
  • Datta, G. S., Mukerjee, R., Ghosh, M. and Sweeting, T. J. (2000). Bayesian prediction with approximate frequentist validity. Ann. Statist. 28 1414--1426.
  • Dawid, A. P. and Sebastiani, P. (1999). Coherent dispersion criteria for optimal experimental design. Ann. Statist. 27 65--81.
  • Dawid, A. P. and Vovk, V. G. (1999). Prequential probability: Principles and properties. Bernoulli 5 125--162.
  • de Finetti, B. (1970). Teoria delle Probabilità 1, 2. Einaudi, Torino. [English translations published (1974, 1975) as Theory of Probability 1, 2. Wiley, New York.]
  • Delampady, M., DasGupta, A., Casella, G., Rubin, H. and Strawderman, W. E. (2001). A new approach to default priors and robust Bayes methodology. Canad. J. Statist. 29 437--450.
  • Dey, D., Müller, P. and Sinha, D., eds. (1998). Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statist. 133. Springer, New York.
  • Diaconis, P. (1988a). Bayesian numerical analysis. In Statistical Decision Theory and Related Topics IV (S. Gupta and J. Berger, eds.) 1 163--175. Springer, New York.
  • Diaconis, P. (1988b). Recent progress on de Finetti's notion of exchangeability. In Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) 111--125. Oxford Univ. Press.
  • Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates (with discussion). Ann. Statist. 14 1--67.
  • Eaton, M. L. (1989). Group Invariance Applications in Statistics. IMS, Hayward, CA.
  • Efron, B. (1993). Bayes and likelihood calculations from confidence intervals. Biometrika 80 3--26.
  • Fraser, D. A. S., Reid, N., Wong, A. and Yi, G. Y. (2003). Direct Bayes for interest parameters. In Bayesian Statistics 7 (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds.) 529--534. Oxford Univ. Press.
  • Freedman, D. A. (1963). On the asymptotic behavior of Bayes estimates in the discrete case. Ann. Math. Statist. 34 1386--1403.
  • Freedman, D. A. (1999). On the Bernstein--von Mises theorem with infinite-dimensional parameters. Ann. Statist. 27 1119--1140.
  • Gart, J. J. and Nam, J. (1988). Approximate interval estimation of the ratio of binomial parameters: A review and corrections for skewness. Biometrics 44 323--338.
  • Gelman, A., Carlin, J. B., Stern, H. and Rubin, D. B. (1995). Bayesian Data Analysis. Chapman and Hall, London.
  • Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500--531.
  • Ghosh, J. K., ed. (1988). Statistical Information and Likelihood. A Collection of Critical Essays. Lecture Notes in Statist. 45. Springer, New York.
  • Ghosh, J. K., Ghosal, S. and Samanta, T. (1994). Stability and convergence of the posterior in non-regular problems. In Statistical Decision Theory and Related Topics V (S. S. Gupta and J. Berger, eds.) 183--199. Springer, New York.
  • Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.
  • Ghosh, M. and Kim, Y.-H. (2001). The Behrens--Fisher problem revisited: A Bayes--frequentist synthesis. Canad. J. Statist. 29 5--17.
  • Good, I. J. (1983). Good Thinking: The Foundations of Probability and Its Applications. Univ. Minnesota Press, Minneapolis.
  • Guttman, I. (1967). The use of the concept of a future observation in goodness-of-fit problems. J. Roy. Statist. Soc. Ser. B 29 83--100.
  • Hill, B. (1974). On coherence, inadmissibility and inference about many parameters in the theory of least squares. In Studies in Bayesian Econometrics and Statistics (S. Fienberg and A. Zellner, eds.) 555--584. North-Holland, Amsterdam.
  • Hobert, J. (2000). Hierarchical models: A current computational perspective. J. Amer. Statist. Assoc. 95 1312--1316.
  • Hwang, J. T., Casella, G., Robert, C., Wells, M. T. and Farrell, R. (1992). Estimation of accuracy in testing. Ann. Statist. 20 490--509.
  • Jeffreys, H. (1961). Theory of Probability, 3rd ed. Oxford Univ. Press.
  • Kiefer, J. (1977). Conditional confidence statements and confidence estimators (with discussion). J. Amer. Statist. Assoc. 72 789--827.
  • Kim, Y. and Lee, J. (2001). On posterior consistency of survival models. Ann. Statist. 29 666--686.
  • Lad, F. (1996). Operational Subjective Statistical Methods: A Mathematical, Philosophical and Historical Introduction. Wiley, New York.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation, 2nd ed. Springer, New York.
  • McDonald, G. C., Vance, L. C. and Gibbons, D. I. (1995). Some tests for discriminating between lognormal and Weibull distributions---an application to emissions data. In Recent Advances in Life-Testing and Reliability---A Volume in Honor of Alonzo Clifford Cohen, Jr. (N. Balakrishnan, ed.) Chapter 25. CRC Press, Boca Raton, FL.
  • Meng, X.-L. (1994). Posterior predictive $p$-values. Ann. Statist. 22 1142--1160.
  • Morris, C. (1983). Parametric empirical Bayes inference: Theory and applications (with discussion). J. Amer. Statist. Assoc. 78 47--65.
  • Mossman, D. and Berger, J. (2001). Intervals for post-test probabilities: A comparison of five methods. Medical Decision Making 21 498--507.
  • Neyman, J. (1977). Frequentist probability and frequentist statistics. Synthèse 36 97--131.
  • Neyman, J. and Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica 16 1--32.
  • O'Hagan, A. (1992). Some Bayesian numerical analysis. In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 345--363. Oxford Univ. Press.
  • Paulo, R. (2002). Problems on the Bayesian/frequentist interface. Ph.D. dissertation, Duke Univ.
  • Pratt, J. W. (1965). Bayesian interpretation of standard inference statements (with discussion). J. Roy. Statist. Soc. Ser. B 27 169--203.
  • Rao, J. N. K. (2003). Small Area Estimation. Wiley, New York.
  • Reid, N. (2000). Likelihood. J. Amer. Statist. Assoc. 95 1335--1340.
  • Ríos Insua, D. and Ruggeri, F., eds. (2000). Robust Bayesian Analysis. Lecture Notes in Statist. 152. Springer, New York.
  • Robbins, H. (1955). An empirical Bayes approach to statistics. Proc. Third Berkeley Symp. Math. Statist. Probab. 1 157--164. Univ. California Press, Berkeley.
  • Robert, C. P. (2001). The Bayesian Choice, 2nd ed. Springer, New York.
  • Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods. Springer, New York.
  • Robins, J. M. and Ritov, Y. (1997). Toward a curse of dimensionality appropriate (CODA) asymptotic theory for semi-parametric models. Statistics in Medicine 16 285--319.
  • Robins, J. M., van der Vaart, A. and Ventura, V. (2000). Asymptotic distribution of $p$-values in composite null models. J. Amer. Statist. Assoc. 95 1143--1156.
  • Robins, J. and Wasserman, L. (2000). Conditioning, likelihood and coherence: A review of some foundational concepts. J. Amer. Statist. Assoc. 95 1340--1346.
  • Robinson, G. K. (1979). Conditional properties of statistical procedures. Ann. Statist. 7 742--755.
  • Rousseau, J. (2000). Coverage properties of one-sided intervals in the discrete case and applications to matching priors. Ann. Inst. Statist. Math. 52 28--42.
  • Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 1151--1172.
  • Rubin, H. (1987). A weak system of axioms for ``rational'' behavior and the non-separability of utility from prior. Statist. Decisions 5 47--58.
  • Savage, L. J. (1962). The Foundations of Statistical Inference. Methuen, London.
  • Schervish, M. (1995). Theory of Statistics. Springer, New York.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461--464.
  • Sellke, T., Bayarri, M. J. and Berger, J. (2001). Calibration of $p$-values for testing precise null hypotheses. Amer. Statist. 55 62--71.
  • Soofi, E. (2000). Principal information theoretic approaches. J. Amer. Statist. Assoc. 95 1349--1353.
  • Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proc. Third Berkeley Symp. Math. Statist. Probab 1 197--206. Univ. California Press, Berkeley.
  • Stein, C. (1975). Estimation of a covariance matrix. Reitz Lecture, IMS--ASA Annual Meeting. (Also unpublished lecture notes.)
  • Strawderman, W. (2000). Minimaxity. J. Amer. Statist. Assoc. 95 1364--1368.
  • Sun, D. and Berger, J. (2003). Objective priors under sequential experimentation. Technical report, Univ. Missouri.
  • Sweeting, T. J. (2001). Coverage probability bias, objective Bayes and the likelihood principle. Biometrika 88 657--675.
  • Tang, D. (2001). Choice of priors for hierarchical models: Admissibility and computation. Ph.D. dissertation, Purdue Univ.
  • Vidakovic, B. (2000). Gamma-minimax: A paradigm for conservative robust Bayesians. In Robust Bayesian Analysis. Lecture Notes in Statist. 152 241--259. Springer, New York.
  • Wald, A. (1950). Statistical Decision Functions. Wiley, New York.
  • Welch, B. and Peers, H. (1963). On formulae for confidence points based on integrals of weighted likelihoods. J. Roy. Statist. Soc. Ser. B 25 318--329.
  • Wolpert, R. L. (1996). Testing simple hypotheses. In Data Analysis and Information Systems: Statistical and Conceptual Approaches (H.-H. Bock and W. Polasek, eds.) 289--297. Springer, Berlin.
  • Woodroofe, M. (1986). Very weak expansions for sequential confidence levels. Ann. Statist. 14 1049--1067.
  • Yang, R. and Berger, J. (1994). Estimation of a covariance matrix using the reference prior. Ann. Statist. 22 1195--1211.
  • Ye, K. (1993). Reference priors when the stopping rule depends on the parameter of interest. J. Amer. Statist. Assoc. 88 360--363.
  • Zhao, L. (2000). Bayesian aspects of some nonparametric problems. Ann. Statist. 28 532--552.