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An overview of robust Bayesian analysis

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Summary

Robust Bayesian analysis is the study of the sensitivity of Bayesian answers to uncertain inputs. This paper seeks to provide an overview of the subject, one that is accessible to statisticians outside the field. Recent developments in the area are also reviewed, though with very uneven emphasis.

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References

  • Angers, J. F. (1992). Use of the student-t prior for the estimation of normal means: a computational approach.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.), Oxford: University Press, 567–576.

    Google Scholar 

  • Angers, J. F. and Berger, J. O. (1991). Robust hierarchical Bayes estimation of exchangeable means.Canadian J. Statist. 19, 39–56.

    MathSciNet  MATH  Google Scholar 

  • Angers, J. F. and Delampady, M. (1992). Hierarchical Bayesian curve fitting and smoothing.Canadian J. Statist. 20, 35–49.

    MathSciNet  MATH  Google Scholar 

  • Angers, J. F., MacGibbon, B. and Wang, S. (1992). A robust Bayesian approach to the estimation of intra-block exchangeable normal means with applications.Tech. Rep. 92-7, Université de Montréal.

  • Basu, S. (1992a). Variations of posterior expectations for symmetric unimodal priors in a distribution band.Tech. Rep. 214, Department of Statistics and Applied Probability, University of California, Santa Barbara.

    Google Scholar 

  • Basu, S. (1992b). Ranges of posterior probability: symmetry, unimodality and the likelihood.Tech. Rep. 215. University of California, Santa Barbara.

    Google Scholar 

  • Basu, S. (1992c). A new look at Bayesian point null hypothesis testing: HPD sets, volume minimizing sets, and robust Bayes.Tech. Rep. 216, University of California, Santa Barbara

    Google Scholar 

  • Basu, S. and DasGupta, A. (1992). Bayesian analysis with distribution bands: the role of the loss function.Tech. Rep. 208. University of California, Santa Barbara.

    Google Scholar 

  • Basu, S. and Jammalamadaka, S. R. (1993). Local posterior robustness with parametric priors: maximum and average sensitivity.Tech. Rep. 239, University of California, Santa Barbara.

    Google Scholar 

  • Basu, S., Jammalamadaka, S. R., and Liu, Wei (1993a). Local posterior robustness: total derivatives and functional derivatives.Tech. Rep. 239, University of California, Santa Barbana.

    Google Scholar 

  • Basu, S. Jammalamadaka, S. R., and Liu, W. (1993b). Qualitative robustness and stability of the posterior distributions and posterior quantities.Tech. Rep. 238, University of California, Santa Barbara

    Google Scholar 

  • Bayarri, M. J. and Berger, J. (1993a). Robust Bayesian analysis of selection models.Tech. Rep. 93-6, Purdue University, W. Lafayette.

    Google Scholar 

  • Bayarri, M. J. and Berger, J. (1993b). Robust Bayesian bounds for outlier detection.Proceedings of the 4th International Meeting of Statistics in the Basque Country-IMSIBAC4 (M. L. Puri and J. P. Vilaplana, eds.). Amsterdam: North-Holland.

    Google Scholar 

  • Bayarri, M. J. and Berger, J. (1994). Applications and limitations of robust Bayesian bounds and type II MLE.Statistical Decision Theory and Related Topics V (S. S. Gupta and J. O. Berger, eds.). Berlin: Springer, 121–134.

    Google Scholar 

  • Bayarri, M. J. and DeGroot, M. (1992). A BAD view of weighted distributions and selection models.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 17–34.

    Google Scholar 

  • Berger, J. (1984). The robust Bayesian viewpoint.Robustness of Bayesian Analysis (J. B. Kadane, ed.). Amsterdam: North-Holland, 63–124, (with discussion).

    Google Scholar 

  • Berger, J. O. (1985).Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.

    MATH  Google Scholar 

  • Berger, J. O. (1990). Robust Bayesian analysis: sensitivity to the prior.J. Statist. Planning and Inference 25, 303–328.

    Article  MATH  Google Scholar 

  • Berger, J. O. (1992). A comparison of minimal Bayesian tests of precise hypotheses.Rassegna di Metodi Statistici ed Applicazioni 7, Pitagora Editrice. Bologna, 43–78

    Google Scholar 

  • Berger, J. O. and Berliner, L. M. (1986). Robust Bayes and empirical Bayes analysis with ∈-contaminated priors.Ann. Statist. 14, 461–486.

    MathSciNet  MATH  Google Scholar 

  • Berger, J. and Bernardo, J. M. (1992). On the development of the reference prior method.Bayesian Statistics 4 J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 35–60.

    Google Scholar 

  • Berger, J. O., Brown, L. and Wolpert, R. (1994). A unified conditional frequentist and Bayesian test for fixed and sequential hypothesis testing.Ann. Statist. (to appear).

  • Berger, J. O. and Chen, M. H. (1993). Determining retirement patterns: prediction for a multinomial distribution with constrained parameter space.The Statistician 42, 427–443.

    Article  Google Scholar 

  • Berger, J. O. and Delampady, M. (1987). Testing precise hypotheses.Statist. Sci. 2, 317–352, (with discussion).

    MathSciNet  MATH  Google Scholar 

  • Berger, J. O. and Jefferys, W. (1992). The application of robust Bayesian analysis to hypothesis testing and Occam’s razor.J. It. Statist. Soc. 1, 17–32.

    MATH  Google Scholar 

  • Berger, J. O. and Moreno, E. (1994). Bayesian robustness in bidimensional models: prior independence.J. Statist. Planning and Inference. (to appear).

  • Berger, J. O. and O’Hagan, A. (1988). Ranges of posterior probabilities for unimodal priors with specified quantiles.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Oxford: University Press, 45–66.

    Google Scholar 

  • Berger, J. O. and Pericchi, L. R. (1993). The intrinsic Bayes factor for model selection and prediction.Tech. Rep. 93-43C, Purdue University, W. Lafayette.

    Google Scholar 

  • Berger, J. O. and Robert, C. (1990). Subjective hierarchical Bayes estimation of a multivariate normal mean: on the frequentist interface.Ann. Statist. 18, 617–651.

    MathSciNet  MATH  Google Scholar 

  • Berger, J. O. and Sellke, T. (1987. Testing a point null hypothesis: the irreconcilability of significance levels and evidence.J. Amer. Statist. Assoc. 82, 112–122, (with discussion).

    Article  MathSciNet  MATH  Google Scholar 

  • Berliner, L. M. and MacEachern, S. N. (1993). Examples of inconsistent Bayes procedures based on observations on dynamical systems.Statistics and Probab. Letters 17, 355–360.

    Article  MathSciNet  MATH  Google Scholar 

  • Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference.J. Roy. Statist. Soc. A 41, 113–147, (with discussion).

    MathSciNet  MATH  Google Scholar 

  • Bernardo, J. M. and Smith, A. F. M. (1994).Bayesian Theory. New York: Wiley.

    MATH  Google Scholar 

  • Berry, D., Wolff, M. C. and Sack, D. (1992). Public health decision making: a sequential vaccine trial.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 79–96.

    Google Scholar 

  • Betrò, B., Meczarski, M., and Ruggeri, F. (1994). Robust Bayesian analysis under generalized moment conditions.J. Statist. Planning and Inference, (to appear).

  • Boratyńska, A. (1991). On bayesian robustness with the ε-contamination class of priors.Tech. Rep., University of Warsaw

  • Boratyńska, A. and Zieliński, R. (1991). Infinitesimal Bayes robustness in the Kolmogorov and the Lévy metrics.Tech. Rep. University of Warsaw

  • Bose, S. (1990).Bayesian Robustness with Shape-constrained Priors and Mixtures of Priors. Ph.D. Thesis, Purdue University.

  • Bose, S. (1993). Bayesian robustness with mixture classes of priors.Tech. Rep. 93-1, George Washington University.

  • Bose, S. (1994). Bayesian robustness with more than one class of contaminations.J. Statist. Planning and Inference, (to appear),

  • Box, G. E. P. (1980). Sampling and Bayes’ inference in scientific modelling and robustness.J. Roy. Statist. Soc. A 143, 383–430, (with discussion).

    MathSciNet  MATH  Google Scholar 

  • Box, G. and Tiao, G. (1973).Bayesian Inference in Statistical Analysis. Reading, MA: Addison-Wesley.

    MATH  Google Scholar 

  • Brunner, L. and Lo, A. (1989). Bayes methods for a symmetric unimodal density and its mode.Ann. Statist. 17, 1550–1566.

    MathSciNet  MATH  Google Scholar 

  • Cano, J. A. (1993). Robustness of the posterior mean in normal hierarchical models.Comm. Statist. Theory and Methods 22, 1999–2014.

    MathSciNet  MATH  Google Scholar 

  • Cano, J. A., Hernández, A. and Moreno, E. (1985). Posterior measures under partial prior information.Statistica 2, 219–230.

    Google Scholar 

  • Carlin, B. P., Chaloner, K. M., Louis, T. A., and Rhame, F. S. (1993). Elicitation, monitoring, and analysis for an AIDS clinical trial.Tech. Rep., University of Minnesota

  • Carlin, B. P. and Louis, T. A. (1993). Identifying prior distributions that produce specific decisions, with application to monitoring clinical trials.Bayesian Inference in Econometrics and Statistics, (to appear).

  • Carlin, B. P., and Polson, N. G. (1991). An expected utility approach to influence diagnostics.J. Amer. Statist. Assoc. 86, 1013–1021.

    Article  Google Scholar 

  • Casella, G. and Berger, R. (1987). Reconciling Bayesian and frequentist evidence in the one-sided testing problem.J. Amer. Statist. Assoc. 82, 106–111 (with discussion).

    Article  MathSciNet  MATH  Google Scholar 

  • Casella G. and Wells, M. (1991). Noninformative priors for robust Bayesian inference.Tech. Rep., Cornell University

  • Chib, S., Osiewalski, J. and Steel, M. F. J. (1991). Posterior inference on the degrees of freedom parameter in multivariate-t regression models.Economics Letters 37, 391–397.

    Article  MathSciNet  MATH  Google Scholar 

  • Coolen, F. P. A. (1993). Imprecise conjugate prior densities for the one-parameter exponential model.Statistics and Probability Letters 16, 337–342.

    Article  MathSciNet  MATH  Google Scholar 

  • Cuevas, A. and Sanz P. (1988). On differentiability properties of Bayes Operators.Bayesian Statistics, 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) Oxford: University Press, 569–577.

    Google Scholar 

  • DasGupta, A. (1991). Diameter and volume minimizing confidence sets in Bayes and classical problems.Ann. Statist. 19, 1225–1243.

    MathSciNet  MATH  Google Scholar 

  • DasGupta, A. and Delampady, M. (1990). Bayesian testing with symmetric and unimodal priors.Tech. Rep. 90-47, Purdue University.

  • DasGupta, A. and Mukhopadhyay, S. (1994). Uniform and subuniform posterior robustness: the sample size problem.J. Statist. Planning and Inference, (to appear).

  • DasGupta, A. and Studden, W.J. (1988a). Frequentist behavior of robust Bayes procedures: new applications of the Lehmann-Wald minimax theory to a novel geometric game.Tech. Rep. 88-36C, Purdue University.

  • DasGupta, A. and Studden, W. J. (1988b). Robust Bayesian analysis in normal linear models with many parameters.Tech. Rep. 88-14, Purdue University.

  • DasGupta, A. and Studden, W.J. (1989). Frequentist behavior of robust Bayes estimates of normal means.Statistics and Decisions,7, 333–361.

    MathSciNet  MATH  Google Scholar 

  • DasGupta, A. and Studden, W. J. (1991). Robust Bayesian experimental designs in normal linear models.Ann. Statist. 19, 1244–1256.

    MathSciNet  MATH  Google Scholar 

  • Datta, G. and Lahiri, P. (1992). Robust hierarchical Bayes estimation of small area characteristics in presence of covariates.Tech. Rep. 92-28, University of Georgia.

  • Dawid, A. P. (1973). Posterior expectations for large observations.Biometrika,60, 664–667.

    Article  MathSciNet  MATH  Google Scholar 

  • De la Horra, J., and Fernandez, C. (1993). Bayesian analysis under ε-contaminated priors: a trade-off between robustness and precision.J. Statist. Planning and Inference 38, 13–30.

    Article  Google Scholar 

  • De la Horra, J. and Fernández, C. (1994). Bayesian robustness of credible regions in the presence of nuisance parameters.Communications in Statistics 23. (to appear).

  • Delampady, M. (1989a). Lower bounds on Bayes factors for interval null hypotheses.J. Amer. Statist. Assoc. 84, 120–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Delampady, M. (1989b). Lower bounds on Bayes factors for invariant testing situations.J. Multivariate Analysis 28, 227–246.

    Article  MathSciNet  MATH  Google Scholar 

  • Delampady, M. and Berger J. (1990). Lower bounds on Bayes factors for Multinomial and chi-squared tests of fit.Ann. Statist. 18, 1295–1316.

    MathSciNet  MATH  Google Scholar 

  • Delampady, M. and Dey, D. (1994). Bayesian robustness for multiparameter problems.J. Statist. Planning and Inference, (to appear).

  • De Robertis, L. (1978). The use of partial prior knowledge in Bayesian inference. Ph.D. Thesis, Yale University.

  • DeRobertis, L. and Hartigan, J. A. (1981). Bayesian inference using intervals of measures.Ann. Statist. 1, 235–244.

    MathSciNet  Google Scholar 

  • Dette, H. and Studden, W. J. (1994). A geometric solution of the BayesianE-optimal design problem.Statistical Decision Theory and Related Topics V (S. S. Gupta and J. O. Berger, eds.), Berlin, Springer, 157–170.

    Google Scholar 

  • Dey, D. and Birmiwal (1991). Robust Bayesian analysis using entropy and divergence measures.Tech. Rep., University of Connecticut

  • Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates.Ann. Statist. 14, 1–67.

    MathSciNet  MATH  Google Scholar 

  • Doss, H. (1994). Bayesian estimation for censored data: an experiment in sensitivity analysis.Statistical Decision Theory and Related Topics, V (S. S. Gupta and J. O. Berger, eds.) Berlin: Springer, 171–182.

    Google Scholar 

  • Draper, D. (1992). Assessment and propagation of model uncertaintyTech. Rep., University of California

  • Drummey, K. W. (1991)Robust Bayesian Estimation in the Normal, Gamma, and Binomial Probability Models: a Computational Approach. Ph.D. Thesis, University of Maryland.

  • DuMouchel, W. and Harris, J. (1983). Bayes methods for combining the results of cancer studies in human other species.J. Amer. Statist. Assoc. 78, 293–315, (with discussion).

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards, W., Lindman, H. and Savage, L. J. (1963). Bayesian statistical inference for psychological research.Psychological Rev. 70, 193–242.

    Article  Google Scholar 

  • Eichenauer-Herrmann, J. and Ickstadt, K. (1993). A saddle point characterization for classes of priors with shape-restricted densities.Statistics and Decisions 11, 175–179.

    MathSciNet  MATH  Google Scholar 

  • Fan, T. H. and Berger, J. O. (1990). Exact convolution oft-distributions, with application to Bayesian inference for a normal mean witht prior distributions.J. Statist. Computation and Simulation 36, 209–228.

    MathSciNet  MATH  Google Scholar 

  • Fan, T. H. and Berger, J. O. (1992). Behavior of the posterior distribution and inferences for a normal mean witht prior distributions.Statistics and Decisions 10, 99–120.

    MathSciNet  MATH  Google Scholar 

  • Ferguson, T.S., Phadia, E.G., and Tiwari, R.C. (1992). Bayesian nonparametric inference.Current Issues in Statistical Inference: Essays in Honor of D. Basu (M. Ghosh, and P.K. Pathak, eds.), Hayward CA.: IMS, 127–150.

    Google Scholar 

  • Fernández, C., Osiewalski, J., and Steel, M.F.J. (1993). Marginal equivalence inv-spherical models.Tech. Rep., Universidad Autónoma de Madrid

  • Fortini, S., and Ruggeri, F. (1990). Concentration function in a robust Bayesian framework.Rech. Rep. 90.6, CNR-IAMI, Milano.

    Google Scholar 

  • Fortini, S. and Ruggeri, F. (1992). On definining neighborhoods of measures through the concentration function.Sankhyā A. (to appear).

  • Fortini, S. and Ruggeri, F. (1994). Concentration functions and Bayesian robustness.J. Statist. Planning and Inference, (to appear).

  • Fougere, P. (Ed.) (1990).Maximum Entropy and Bayesian Methods. Dordrecht: Kluwer

    MATH  Google Scholar 

  • Gasparini, M. (1990). Nonparametric Bayes estimation of a distribution function with truncated data.Tech. Rep. 182, University of Michigan.

  • Geisser, S. (1992). Bayesian perturbation diagnostics and robustness.Bayesian Analysis in Statistics and Econometrics (P. K. Goel and N. S. Iyengar, eds.). Berlin: Springer, 289–302.

    Google Scholar 

  • Gelfand, A. and Dey, D. (1991). On Bayesian robustness of contaminated classes of priors.Statitics and Decisions 9, 63–80.

    MathSciNet  MATH  Google Scholar 

  • Genest, C., and Zidek, J. (1986). Combining probability distributions: A critique and an annotated bibliography.Statist. Sci. 1, 114–135.

    MathSciNet  Google Scholar 

  • Geweke, J. (1992). Priors for macroeconomic time series and their application.Proceedings of the Conference on Bayes Methods and Unit Roots, a special issue ofEconometric Theory, (to appear).

  • Ghosh, J. K., Ghosal, S., and Samanta, T. (1994). Stability and convergence of the posterior in non-regular problems.Statistical Decision Theory and Related Topics V (S. S. Gupta and J. O. Berger, eds.). Berlin: Springer, 183–200.

    Google Scholar 

  • Ghosh, M. (1993). Inconsistent maximum likelihood estimators for the Rasch model.Tech. Rep., University of Florida.

  • Ghosh, M. (1994). On some Bayesian solutions of the Neyman-Scott problem.Statistical Decision Theory and Related Topics V (S. S. Gupta and J.O. Berger, eds.). Berlin: Springer, 267–276.

    Google Scholar 

  • Girón, F.J. and Ríos, S. (1980). Quasi-Bayesian behavior: a more realistic approach to decision making?Bayesian Statistics (J.M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Valencia: University Press, 17–38.

    Google Scholar 

  • Goldstein, M. and Wooff, D.A. (1994). Robustness measures for Bayes linear analyses.J. Statist. Planning and Inference, (to appear).

  • Good, I. J. (1983a).Good Thinking: The Foundations of Proability and Its Applications. University of Minnesota Press, Minneapolis, MN.

    Google Scholar 

  • Good, I. J. (1983b). The robustness of a hierarchical model for multinomials and contingency tables.Scientific Inference, Data Analysis and Robustness (G. E. P. Box, T. Leonard and C. F. Wu, eds.). New York: Academic Press.

    Google Scholar 

  • Good, I. J. and Crook, J. F. (1987). The robustness and sensitivity of the mixedi-Dirichlet Bayesian test for ‘independence’ in contingency tables.Ann. Statist. 15, 694–711.

    MathSciNet  MATH  Google Scholar 

  • Goutis, C. (1991). Ranges of posterior measures for some classes of priors with specified moments.Tech. Rep. 70, University College London.

  • Gu, C. and Wahba, G. (1993). Smoothing spline ANOVA with component-wisa Bayesian confidence intervals.J. Comput. and Graphical Stat. 2, 97–117.

    Article  MathSciNet  Google Scholar 

  • Gustafson, P. and Wasserman, L. (1993). Local sensitivity diagnostics for Bayesian inference.Tech. Rep. 574, Carnegie-Mellon University.

  • Guttman, I. and Peña, D. (1988). Outliers and influence: evaluation by posteriors of parameters in the linear model.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Oxford: University Press, 631–640.

    Google Scholar 

  • Guttman, I. and Peña, D. (1993). A Bayesian look at diagnostics in the univariate linear model.Statistica Sinica 3, 367–390.

    MathSciNet  MATH  Google Scholar 

  • Hartigan, J.A. (1983).Bayes Theory, New York: Springer-Verlag.

    MATH  Google Scholar 

  • Ickstadt, K. (1992). Gamma-minimax estimators with respect to unimodal priors. InIperations Research ’91 (P. Gritzmann,et al., eds.), Heidelberg: Physica-Verlag.

    Google Scholar 

  • Jaynes, E. T. (1983).Papers on Probability, Statistics and Statistical Physics, (R. Rosenkrantz, ed.), Dordrecht: Reidel

    Google Scholar 

  • Jefferys, W., Berger, J. (1992). Ockham’s razor and Bayesian analysis.Amer. Scientist 80, 64–72.

    Google Scholar 

  • Kadane, J. (1994). An application of robust Bayesian analysis to a medical experiment.J. Statist. Planning and Inference, (to appear).

  • Kadane, J. B. and Chuang, D. T. (1978). Stable decision problems.Ann. Statist. 6, 1095–1110.

    MathSciNet  MATH  Google Scholar 

  • Kass, R. and Greenhouse, J. (1989). Investigating therapies of potentially great benefit: A Bayesian perspective. Comments on “Investigating therapies of potentially great benefit: ECMO”, by J. H. Ware.Statist. Sci. 4, 310–317.

    Google Scholar 

  • Kass, R. E. and Raftery, A. (1992). Bayes factors and model uncertainty.Tech. Rep. 571, Carnegic-Mellon University.

  • Kass, R. E., and Slate, E. H. (1992). Reparametrization and diagnostics of posterior non-normality.Bayesian Statitics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 289–306.

    Google Scholar 

  • Kass, R. E., Tierney, L. and Kadane, J. B. (1980). Approximate methods for assessing influence and sensitivity in Bayesian analysis.Biometrika 76, 663–674.

    Article  MathSciNet  Google Scholar 

  • Kass, R. E. and Wasserman, L. (1993). Formal rules of selecting prior distributions: a review and annotated bibliography.Tech. Rep., Carnegie-Mellon University.

  • Kiefer, J. (1977). Conditional confidence statements and confidence estimators (theory and methods).J. Amer. Statist. Assoc. 72, 789–827.

    Article  MathSciNet  MATH  Google Scholar 

  • Kohn, R. and Ansley, C. F. (1988). The equivalence between Bayesian smoothness priors and optimal smoothing, for function estimation. InBayesian Analysis of Time Series and Dynamic Models (J. C. Spall, ed.). New York: Marcel Dekker.

    Google Scholar 

  • Kouznetsov, V. P. (1991).Interval Statistical Models. Moscow: Radio and Communication.

    Google Scholar 

  • Laplace, P. S. (1812).Theorie Analytique des Probabilities. Courcier, Paris.

    Google Scholar 

  • Lavine, M. (1989). The boon of dimensionality: how to be a sensitive multidimensional Bayesian.Tech. Rep. 89-14, Duke University.

  • Lavine, M. (1991a). Sensitivity in Bayesian statistics: the prior and the likelihood.J. Amer. Statist. Assoc. 86, 396–399.

    Article  MathSciNet  MATH  Google Scholar 

  • Levine, M. (1991b). An approach to robust Bayesian analysis with multidimensional spaces.J. Amer. Statist. Assoc. 86, 400–403.

    Article  MathSciNet  Google Scholar 

  • Lavine, M. (1992a). Sensitivity in Bayesian statistics: the prior and the likelihood.J. Ame. Statist. Assoc. 86, 396–399.

    Article  MathSciNet  Google Scholar 

  • Lavine, M. (1992b). Some aspects of Polya tree distributions for statistical modelling.Ann. Statist. 20, 1222–1235.

    MathSciNet  MATH  Google Scholar 

  • Lavine, M. (1992c). A note on bounding Monte Carlo variances.Comm. Statist. Theory and Methods 21, 2855–2860.

    MathSciNet  MATH  Google Scholar 

  • Lavine, M. (1992d). Local predictive influence in Bayesian linear models with conjugate priors.Commun. Statist.-Simulation 21, 269–283.

    MathSciNet  MATH  Google Scholar 

  • Lavine, M. (1994). An approach to evaluating sensitivity in Bayesian regression analysis.J. Statist. Planning and Inference. (to appear).

  • Lavine, M. and Wasserman, L. (1992). Can we estimateN? Tech. Rep. 546, Carnegie-Mellon University.

  • Lavine, M., Wasserman, L. and Wolpert, R. (1991). Bayesian inference with specified prior marginals.J. Amer. Statist. Assoc. 86, 964–971.

    Article  MathSciNet  MATH  Google Scholar 

  • Lavine, M., Wasserman, L., and Wolpert, R. (1993). Linearization of Bayesian robustness problems.J. Statist. Planning and Inference 37, 307–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Leamer, E. E. (1982). Sets of posterior means with bounded variance prior.Econometrica 50, 725–736.

    Article  MathSciNet  MATH  Google Scholar 

  • Lenk, P. J. (1988). The logistic normal distribution for Bayesian, nonparametric, predictive densities.J. Amer. Statist. Assoc. 83, 509–516.

    Article  MathSciNet  MATH  Google Scholar 

  • Leonard T. (1978). Density estimation, stochastic processes, and prior information.J. Roy. Statist. Soc. B 40, 113–146.

    MathSciNet  MATH  Google Scholar 

  • Li, Y. and Saxena, K. M. L. (1990). Optimal robust Bayesian estimation.Tech. Rep. 4, University of Nebraska.

  • Liseo, B. (1993). Elimination of nuisance parameters with reference priors.Biometrika 80, 295–304.

    Article  MathSciNet  MATH  Google Scholar 

  • Liseo, B., Petrella, L., and Salinetti, G. (1993). Block unimodality for multivariate Bayesian robustness.J. Ital. Statist. Soc. (to appear).

  • Lo, A. Y. and Weng, C. S. (1989). On a class of Bayesian nonparametric estimates: II. Hazard rate estimates.Ann. Inst. Statist. Math. 41, 227–245.

    Article  MathSciNet  MATH  Google Scholar 

  • Lucas, T. W. (1992). When is conflict normal?Tech. Rep., The Rand Corporation.

  • Meczarski, M. (1991). Stable Bayesian estimation in the Poisson model: a nonlinear problem.Tech. Rep. 91.14, CNR-IAMI, Milano.

    Google Scholar 

  • Meczarski, M. and Zieliński, R. (1991). Stability of the Bayesian estimator of the Poisson mean under the inexactly specified gamma prior.Statist. Prob. Letters 12, 329–333.

    Article  Google Scholar 

  • Meng, X. L. (1994). Posterior predictiveP-values.Ann. Statist., (to appear).

  • McCulloch, R. E. (1989). Local model influence.J. Amer. Statist. Assoc. 84, 473–478.

    Article  Google Scholar 

  • Moreno, E. and Cano, J. A. (1989). Testing a point null hypothesis: Asymptotic robust Bayesian analysis with respect to the priors given on a subsigma field.Int. Statist. Rev., 57, 221–232.

    MATH  Google Scholar 

  • Moreno, E. and Cano, J. A. (1991). Robust Bayesian analysis for ε-contaminations partially known.J. Roy. Statist. Soc. B 53, 143–155.

    MathSciNet  MATH  Google Scholar 

  • Moreno, E. and Cano, J. A. (1992). Classes of bidimensional priors specified on a collection of sets: Bayesian robustness.Tech. Rep., Universidad de Granada.

  • Moreno, E. and Pericchi, L. R. (1990). Sensitivity of the Bayesian analysis to the priors: structural contaminations with specified quantiles of parametric families.Actas III Cong. Latinoamericano Probab. Estad. Mat., 143–158.

  • Moreno, E. and Pericchi, L. R. (1991). Robust Bayesian analysis for ε-contaminations with shape and quantile constraints.Proc. Fifth Inter. Symp. on Applied Stochastic Models and Data Analysis. World Scientific Publ., 454–470.

  • Moreno, E. and Pericchi, L. R. (1992a). Subjetivismo sin dogmatismo: análisis Bayesiano robusto (with discussion).Estadist. Española 34, 5–60.

    MathSciNet  Google Scholar 

  • Moreno, E. and Pericchi, L. R. (1992b). Bands of probability measures: A robust Bayesian analysis.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.), Oxford: University Press, 707–714.

    Google Scholar 

  • Moreno, E. and Pericchi, L. R. (1993a). Bayesian robustness for hierarchical ε-contamination models.J. Statist. Planning and Inference 37, 159–168.

    Article  MathSciNet  MATH  Google Scholar 

  • Moreno, E. and Pericchi, L. R. (1993b). Precise measurement theory: robust Bayesian analysis.Tech. Rep., Universidad de Granada.

  • Morris, C. N. and Normand, S. L. (1992). Hierarchical models for combining information and for meta-analyses.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 321–344.

    Google Scholar 

  • Moskowitz, H. (1992). Multiple-criteria robust interactive decision analysis for optimizing public policies.Eur. J. Oper. Res. 56, 219–236.

    Article  MATH  Google Scholar 

  • Mukhopadhyay, S. and DasGupta, A. (1993). Uniform approximation of Bayes solutions and posteriors: frequentistly valid Bayes inference.Tech. Rep. 93-12C, Purdue University.

  • O’Hagan, A. (1988). Modelling with heavy tails.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.) Oxford: University Press, 345–360.

    Google Scholar 

  • O’Hagan, A. (1990). Outliers and credence for location parameter inference.J. Amer. Statist. Assoc. 85, 172–176.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Hagan, A. (1994). Robust modelling for asset management.J. Statist. Planning and Inference, (to appear).

  • O’Hagan, A. and Berger, J. O. (1988). Ranges of posterior probabilities for quasi-unimodal priors with specified quantiles.J. Amer. Statist. Assoc. 83, 503–508.

    Article  MathSciNet  MATH  Google Scholar 

  • Osiewalski, J. and Steel, M. F. J. (1993a). Robust Bayesian inference in ℓ q -spherical models.Biometrika, (to appear).

  • Osiewalski, J. and Steel, M. F. J. (1993b). Robust Bayesian inference in elliptical regression models.J. Econometrics 57, 345–363.

    Article  MathSciNet  MATH  Google Scholar 

  • Osiewlaski, J. and Steel, M. F. J. (1993c). Bayesian marginal equivalence of elliptical regression models.Journal of Econometrics, (to appear).

  • Peña, D. and Guttman, I. (1993). Comparing probabilistic methods for outlier detection in linear models.Biometrika 80, 603–610.

    MathSciNet  MATH  Google Scholar 

  • Peña, D. and Tiao, G. C. (1992). Bayesian outlier functions for linear models.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 365–388.

    Google Scholar 

  • Pericchi, L. R. and Nazaret, W. (1988). On being imprecise at the higher levels of a hierarchical linear model.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Oxford: University Press, 361–376.

    Google Scholar 

  • Pericchi, L. R. and Pérez, M. E. (1994). Posterior robustness with more than one sampling model.J. Statist. Planning and Inference., (to appear).

  • Pericchi, L. R. and Walley P. (1991). Robust Bayesian credible intervals and prior ignorance.Internat. Statist. Rev. 58, 1–23.

    Article  Google Scholar 

  • Pettit, L. I. (1988). Bayes methods for outliers in exponential samples.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 731–740.

    Google Scholar 

  • Pettit, L. (1992). Bayes factors for outlier models using the device of imaginary observations.J. Amer. Statist. Assoc. 87, 541–545.

    Article  Google Scholar 

  • Poirier, D. J. (1988). Bayesian diagnostic testing in the general linear normal regression model.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Oxford: University Press, 725–732.

    Google Scholar 

  • Polasek, W. (1985). Sensitivity analysis for general and hierarchical linear regression models.Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti (P. K. Goel and A. Zellner, eds.). Amsterdam: North-Holland

    Google Scholar 

  • Polasek, W. and Pötzelberger, K. (1988). Robust Bayesian analysis in hierarchical models.Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Oxford: University Press, 377–394.

    Google Scholar 

  • Polasek, W. and Pötzelberger, K. (1994). Robust Bayesian methods in simple ANOVA models.J. Statist. Planning and Inference, (to appear).

  • Pötzelberger, K. and Polasek, W. (1991). Robust HPD-regions in Bayesian regression models.Econometrica 59, 1581–1590.

    Article  MathSciNet  MATH  Google Scholar 

  • Rao, C. R. (1985). Weighted distributions arising out of methods of ascertainment: what population does a sample represent?A Celebration of Statistics: The ISI Centenary Volume (A. G. Atkinson and S. Fienberg, eds.). Berlin: Springer.

    Google Scholar 

  • Regazzini, E. (1992). Concentration comparisons between probability measures.Sankhyā B 54, 129–149.

    MathSciNet  MATH  Google Scholar 

  • Ríos-Insúa, D. (1990).Sensitivity Analysis in Multiobjective Decision Making. Berlin: Springer.

    MATH  Google Scholar 

  • Ríos-Insúa, D. (1992). Foundations for a robust theory of decision making: the simple case.Test 1, 69–78.

    MathSciNet  Google Scholar 

  • Ríos-Insúa, D. and French, S. (1991). A framework for sensitivity analysis in discrete multiobjective decision making.Eur. J. Oper. Res. 54, 176–190.

    Article  MATH  Google Scholar 

  • Ríos-Insúa, D. and Martín, J. (1994). Robustness issues under precise beliefs and preferences.J. Statist. Planning and Inference, (to appear).

  • Robert, C. (1992).L’Analyse Statistique Bayesienne. Paris: Economica.

    MATH  Google Scholar 

  • Ruggeri, F. (1990). Posterior ranges of functions of parameters under priors with specified quantites.Comm. Statist. Theory and Methods 19, 127–144.

    MathSciNet  MATH  Google Scholar 

  • Ruggeri, F. (1991). Robust Bayesian analysis given a lower bound on the probability of a set.Comm. Statist. Theory and Methods 20, 1881–1891.

    MathSciNet  MATH  Google Scholar 

  • Ruggeri, F. (1992). Bounds on the prior probability of a set and robust Bayesian analysis.Theory of Probability and Its Applications 37.

  • Ruggeri, F. and Wasserman, L. (1991). Density based classes of priors: Infinitesimal properties and approximations.Tech. Rep. 528, Carnegie-Mellon University.

  • Ruggeri, F. and Wasserman, L. (1993). Infinitesimal sensitivity of posterior distributions.Canadian J. Statist. 21, 195–203.

    MathSciNet  MATH  Google Scholar 

  • Salinetti, G. (1994). Stability of Bayesian decisions.J. Statist. Planning and Inference, (to appear).

  • Sansó, B. and Pericchi, L. R. (1992). Near ignorance classes of log-concave priors for the location model.Test 1, 39–46.

    MathSciNet  Google Scholar 

  • Sedransk, N. (1993). Admissibility of treatment.Clinical Trials: Bayesian Methods and Ethics (J. Kadane ed.). New York: Wiley.

    Google Scholar 

  • Sivaganesan, S. (1988). Range of posterior measures for priors with arbitrary contaminations.Comm. Statst. 17, 1591–1612.

    Article  MATH  Google Scholar 

  • Sivaganesan, S. (1989). Sensitivity of posterior mean to unimodality preserving contaminations.Statistics and Decisions 7, 77–93.

    MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S. (1990). Sensitivity of some standard Bayesian estimates to prior uncertainty—a comparison.J. Statist. Planning and Inference 27, 85–103.

    Article  MathSciNet  Google Scholar 

  • Sivaganesan, S. (1991). Sensitivity of some posterior summaries when the prior is unimodal with specified quantiles.Canadian J. Statist. 19, 57–65.

    MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S. (1992). An evaluation of robustness in binomial empirical Bayes testing.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 783–790.

    Google Scholar 

  • Sivaganesan, S. (1993a). Range of the posterior probability of an interval for priors with unimodality preserving contaminations.Ann. Inst. of Statist. Math. 45, 187–199.

    Article  MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S. (1993b). Optimal robust sets for a density bounded class.Tech. Rep., University of Cincinnati.

  • Sivaganesan, S. (1993c). Robust Bayesian diagnostics.J. Statist. Planning and Inference 35, 171–188.

    Article  MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S. (1994). Bounds on posterior expectations for density bounded classes with constant bandwidth.J. Statist. Planning and Inference., (to appear).

  • Sivaganesan, S. and Berger, J. (1989). Ranges of posterior measures for priors with unimodal contaminations:Ann. Statist. 17, 868–889.

    MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S. and Berger, J. (1993). Robust Bayesian analysis of the binomial empirical Bayes problem.Canadian J. Statist. 21, 107–119.

    MathSciNet  MATH  Google Scholar 

  • Sivaganesan, S., Berliner, L. M., and Berger, J. (1993). Optimal robust credible sets for contaminated priors.Statist. and Probab. Letters, (to appear).

  • Smith, A. F. M. (1983). Bayesian approaches to outliers and robustness.Specifying Statistical Models (J. P. Florens et al. eds.). Berlin: Springer.

    Google Scholar 

  • Spiegelhalter, D. J. (1985). Exact Bayesian inference on the parameters of a Cauchy distribution with vague prior information.Bayesian Statistics 2 (J. M. Bernardo, M. H. DcGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 743–749.

    Google Scholar 

  • Srinivasan, C. and Trusczcynska, H. (1990). Approximation to the range of a ratio linear posterior quantity based on Frechet derivative.Tech. Rep. 289, University of Kentucky.

  • Srinivasan, C. and Trusczcynska, H. (1993). Ranges of non-linear posterior quantities.Ann. Statist., (to appear).

  • Stephens, D. A. and Smith, A. F. M. (1992). Sampling-resampling techniques for the computation of posterior densities in normal means problems.Test 1, 1–18.

    MathSciNet  MATH  Google Scholar 

  • Tamura, H. (1992). Robust Bayesian auditing.Tech. Rep., University of Washington.

  • VanEeden, C. and Zidek, J. V. (1994). Group Bayes estimation of the exponential mean: a retrospective view of the Wald theory.Statistical Decision Theory and Related Topics V (S. S. Gupta and J. O. Berger, eds.). Berlin: Springer, 35–50.

    Google Scholar 

  • Verdinelli, I. and Wasserman, L. (1991). Bayesian analysis of outlier problems using the Gibbs sampler.Statist. Computing 1, 105–117.

    Article  Google Scholar 

  • Vidakovic, B. (1992). A study of properties of computationally simple rules in estimation problems. Ph.D. Thesis, Purdue University.

  • Wahba, G. (1990).Spline Models for Observational Data. Philadelphia: SIAM.

    MATH  Google Scholar 

  • Walley, P. (1991).Statistical Reasoning With Imprecise Probabilities. London: Chapman and Hall.

    MATH  Google Scholar 

  • Wasserman, L. (1989). A robust Bayesian interpretation of likelihood regions.Ann. Statist. 17, 1387–1393.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. (1990). Prior envelopes based on belief functions.Ann. Statist. 18, 454–464.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. (1992a). The conflict between improper priors and robustness.Tech. Rep. 559, Carnegie-Mellon University.

  • Wasserman, L. (1992b). Recent methodological advances in robust Bayesian inference.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 483–502.

    Google Scholar 

  • Wasserman, L. (1992c). Invariance properties of density ratio priors.Ann. Statist. 20, 2177–2182.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. and Kadane, J. (1990). Bayes’ theorem for Choquet capacities.Ann. Statist. 18, 1328–1339.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. and Kadane, J. B. (1992a). Computing bounds on expectation.J. Amer. Statist. Assoc. 87, 516–522.

    Article  MathSciNet  MATH  Google Scholar 

  • Wasserman, L. and Kadane, J. (1992b). Symmetric upper probabilities.Ann. Statist. 20, 1720–1736.

    MathSciNet  MATH  Google Scholar 

  • Wasserman, L. and Seidenfeld, T. (1994). The dilation phenomenon in robust Bayesian inference.J. Statist. Planning and Inference, (to appear).

  • Weiss, R. E. (1992). Influence diagnostics with the Gibbs sampler.Computing Science and Statistics: Proceedings of the 24 th Symposium on the Interface (H. J. Newton, ed.) Alexandria: ASA.

    Google Scholar 

  • Weiss, R. (1993). Identification of outlying perturbations.Tech. Rep. 594, University of Minnesota.

  • West, M. (1987). On scale mixtures of normal distributions.Biometrika 74, 646–648.

    Article  MathSciNet  MATH  Google Scholar 

  • West, M. (1992). Modeling with mixtures.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 503–524.

    Google Scholar 

  • West, M. and Harrison, J. (1989).Bayesian Forecasting and Dynamic Models. Berlin: Springer.

    MATH  Google Scholar 

  • Wolpert, R. L. and Warren Hicks, W. J. (1992). Bayesian hierarchical logistic models for combining field and laboratory survival data.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) Oxford: University Press, 525–546.

    Google Scholar 

  • Zellner, A. (1976). Bayesian and non-Bayesian analysis of regression models with multivariate student-t error terms.J. Amer. Statist. Assoc. 71, 400–405.

    Article  MathSciNet  MATH  Google Scholar 

  • Zen, M. M. and DasGupta, A. (1993). Estimating a binomial parameter: is robust Bayes real Bayes?Statistics and Decisions 11, 37–60.

    MathSciNet  MATH  Google Scholar 

  • Zidek, J. V. and Weerahandi, S. (1992). Bayesian predictive inference for samples from smooth processes.Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 547–566.

    Google Scholar 

Additional References in the Discussion

  • Berger, J. O. (1985b). Discussion of ‘Quantifying prior opinion’ by Diaconis and Ylvisaker.Bayesian Statistics 2 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.), Amsterdam: North-Holland, 133–156.

    Google Scholar 

  • Cano, J.A. and Moreno, E. (1993). Sampling models: a robust Bayesian analysis.Tech. Rep., Universidad de Murcia.

  • Cifarelli, D. M. and Regazzini, E. (1987). On a general definition of concentration function.Sankhyā B 49 307–319.

    MathSciNet  MATH  Google Scholar 

  • Cook, R.D. (1986). Assessment of local influence.J. Roy. Statist. Soc. B 48, 133–169, (with discussion).

    MATH  Google Scholar 

  • Dalal, S. R. and Hall, W. J. (1983). Approximating priors by mixtures of natural conjugate priors.J. Roy. Statist. Soc. B 45, 278–286.

    MathSciNet  MATH  Google Scholar 

  • De la Horra, J. and Fernández, C. (1994). Sensitivity to prior independence via Farlie-Gumbel-Morgenstern model.Tech. Rep. 94-17, Tilburg University.

  • Diaconis, P. and Ylvisaker, D. (1985). Quantifying prior opinion.Bayesian Statistics 2 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds., Amsterdam: North-Holland, 163–175.

    Google Scholar 

  • Efron, B. (1986). Why Isn’t Everyone a Bayesian?Ann. Statist. 40, 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  • Feller, W. (1973).An Introduction to Probability Theory and Applications 2, New York: Wiley.

    Google Scholar 

  • Garthwaite, P. H. (1992). Preposterior expected loss as a scoring rule for prior distributions.Comm. Statist. Th. Meth. 21, 3601–3619.

    MathSciNet  MATH  Google Scholar 

  • Garthwaite, P. H. and Dickey, J. M. (1992). Elicitation of prior distributions for variable-selection problems in regression.Ann. Statist. 20, 1697–1719.

    MathSciNet  MATH  Google Scholar 

  • Grieve, A. P. (1985). A Bayesian Analysis of the Two-Period Crossover Design for Clinical Trials.Biometrika 41, 979–990.

    MathSciNet  Google Scholar 

  • Gustafson, P. (1993a). Local sensitivity in Bayesian statistics. Ph.D. Thesis. Carnegie Mellon University, Pittsburgh.

    Google Scholar 

  • Gustafson, P. (1993b). The local sensitivity of posterior expectations.Tech. Rep., Carnegie Mellon University.

  • Johnson, N. L. and Kotz, S. (1975). On some generalized Farlie-Gumbel-Morgenstern distributions.Comm. Statist. A 4, 415–427.

    Article  MathSciNet  Google Scholar 

  • Johnson, N. L. and Kotz, S. (1977). On some generalized Farlie-Gumbel-Morgenstern distributions II: regression, correlation and further generalizations.Comm. Statist. A 6, 485–496.

    MathSciNet  Google Scholar 

  • Keeney, R. and Raiffa, H. (1993).Decision Making with Multiple Objectives, Wiley.

  • Kemperman, J. H. B. (1972). On a class of moment problem.Proc. Berkeley Symposium on Math. Statist. and Prob. 2, 101–126.

    MathSciNet  Google Scholar 

  • Kemperman, J. H. B. (1983). On the role of duality on the theory of moments.Semiinfinite programming and applications (Fiacco, A. V. and Kortaneck, K. O., eds.). Lecture Notes in Economics and mathematical Systems215, New York: Springer-Verlag.

    Google Scholar 

  • Kemperman, J. H. B. (1987). Geometry of the moment problem.Proceedings of Symposia in Applied mathematics 37, 16–53

    MathSciNet  Google Scholar 

  • Krause, A. and Polasek, W. (1992). Approaches to Tobit Models via Gibbs sampling, in: COMPSTAT 1992, Physica Verlalg, 559–564.

  • Le, H. and O’Hagan, A. (1993). A class of bivariate heavy-tailed distributions.Tech. Rep. 93-20, Nottingham University Statistics Group.

  • Leamer, E. E. (1978). Specification Searches. New York: Wiley.

    MATH  Google Scholar 

  • Leland, J. (1992). An approximate Expected Utility Theory,Tech. Rep., Carnegie Mellon University.

  • Levy, H. (1992). Stochastic Dominance and Expected Utility Analysis: A Review,Mgt. Science 38, 555–593.

    Article  MATH  Google Scholar 

  • Moreno, E., Martínez, C. and Cano, J. A. (1993). Elicitation of contamination classes of prior distributions.Tech. Rep., Universidad de Granada.

  • Nair, V.J., and Wang, P.C.C. (1989). Maximum likelihood estimation under a successive sampling discovery model.Technometrics 31, 423–436.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Hagan, A. and Le, H. (1993). Conflicting information and a class of bivariate heavytailed distributions.Aspects of Uncertainty: A Tribute D. V. Lindley (Smith, A. F. M. and Freeman, P. R., eds.). Wiley: Chichester.

    Google Scholar 

  • Polasek, W. (1992). Joint sensitivity analysis for covariance matrices in Bayesian linear regression.Tech. Rep. 9205, University of Basel.

  • Polasek, W. (1993). Bayesian Generalized Errors in Variables (GEIV) models for censored regression.Tech. Rep., University of Basel.

  • Proll, L., Ríos-Insúa, D. and Salhi, A. (1993). Mathematical Programming and the sensitivity of multicriteria decisions,Annals of Operations Research 43, 109–122.

    Article  MathSciNet  MATH  Google Scholar 

  • Renyi, A. (1962/1970).Wahrscheinlichkeitsrechnung. Berlin: Deutscher Verlag der Wissenschaften. English translation in 1970 asProbability Theory. San Francisco, CA: Holden-Day.

    MATH  Google Scholar 

  • Ríos, S., Ríos-Insúa, S., Ríos-Insúa, D. and Pachón, J. (1994). Experiments in robust decison making (S. Ríos, ed.)Decision Making and Decision Analysis: Trends and Challenges, Dorfrecht: Kluwer, 233–242.

    Google Scholar 

  • Ríos-Insúa, D. (1994). Ambiguity, imprecision and sensitivity in Decison Theory, (Puri and Vilaplana, eds.)New Progress in Probability and Statistics, STP. (to appear).

  • Robert, C. P. (1992).L’Analyse Statistique Bayesianne Paris: Economica.

    Google Scholar 

  • Robert, C. P. (1993). Intrinsic losses. Rapport techn., URA CNRS 1378, Univ. de Rouen.

  • Seidenfeld, T. and Wasserman, L. (1993). Dilation for sets of probabilities.Ann. Statist. 21, 1134–1154.

    MathSciNet  Google Scholar 

  • Stein, C. (1985). On the coverage probability of confidence sets based on a prior distribution. InSequential Methods in Statistics, Banach Center Publication16, 485–514, Warsaw: PWN Publishers.

    Google Scholar 

  • Tibshirani, R. (1989). Noninformative priors for one parameter of many.Biometrika 76, 604–608.

    Article  MathSciNet  MATH  Google Scholar 

  • West, M. (1993). Prediction for finite populations under biased sampling.ISDS Discussion Paper 93, Duke University.

  • Young, N. (1988). Hilbert Space, Cambrdige: University Press.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Purdue University, 47907-1399, West Lafayette, IN, USA

    James O. Berger

  2. Universidad de Granada, Granada, Spain

    Elías Moreno

  3. Universidad Simon Bolivar, Spain

    Luis Raul Pericchi

  4. Universitat de València, València, Spain

    M. Jesús Bayarri & José M. Bernardo

  5. Universidad de Murcia, Murcia, Spain

    Juan A. Cano

  6. Universidad Autónoma de Madrid, Madrid, Spain

    Julián De la Horra

  7. Universidad Politécnica de Madrid, Madrid, Spain

    Jacinto Martín & David Ríos-Insúa

  8. CNR-IAMI, Milano

    Bruno Betrò

  9. Purdue University, USA

    A. Dasgupta

  10. University of Nottingham, Nottingham, UK

    Anthony O’Hagan

  11. University of Basel, Basel, Switzerland

    Wolfgang Polasek

  12. Université de Rouen, Rouen, France

    Christian P. Robert

  13. University College, London

    Constantinos Goutis

  14. Università di Roma “La Sapienza”, Roma, Italy

    Gabriella Salinetti

  15. University of Cincinnati, Cincinnati, USA

    Siva Sivaganesan

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  1. James O. Berger
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Additional information

Carnegie Mellon University

The research of Professor Kadane was supported in part by the following grants; ONR: N0004-89-J-1851, NSF: SES-9123370, DMS-9005858 and DMS-9302557. Professor Srinivasan was supported in part by NSF grants ATM-9108177 and DMS-9204380.

Duke University

CNR-IAMI and Duke University

Read before the Spanish Statistical Society at a meeting organized by the Universidad Autónoma de Madrid on Friday, November 19, 1993

Research supported by the National Science Foundation, Grants DMS-8923071 and DMS 93-03556.

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Berger, J.O., Moreno, E., Pericchi, L.R. et al. An overview of robust Bayesian analysis. Test 3, 5–124 (1994). https://doi.org/10.1007/BF02562676

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  • DOI: https://doi.org/10.1007/BF02562676

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