Consensus priors for multinomial and binomial ratios

References

• Abramowitz, M., and I. A. Stegun. 1964. Handbook of mathematical functions. New York, NY: Dover.
   Google Scholar
• Agresti, A., and Y. Min. 2005. Frequentist performance of Bayesian confidence intervals for comparing proportions in 2 × 2 contingency tables. Biometrics 61:515–23.
   PubMed Web of Science ®Google Scholar
• Aitchison, J., and J. Bacon-Shone. 1981. Bayesian risk ratio analysis. American Statistician 35 (4):254–57.
   Web of Science ®Google Scholar
• Aitkin, M., R. J. Boys, and T. Chadwick. 2005. Bayesian point null hypothesis testing via the posterior likelihood ratio. Statistics and Computing 15:217–30.
   Web of Science ®Google Scholar
• Altham, P. M. E. 1969. Exact Bayesian analysis of a 2 × 2 contingency table, and Fisher’s “exact” significance test. Journal of the Royal Statistical Society, Series B 31 (2):261–69.
   Google Scholar
• Angus, J. E., and R. E. Schafer. 1984. Improved onfidence statements for the binomial parameter. American Statistician 38 (3):189–91.
   Web of Science ®Google Scholar
• Bayes, T. R. 1763. An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society, London 53:370–418.
   Google Scholar
• Berger, J. O., B. Liseo, and R. L. Wolpert. 1999. Integrated likelihood methods for eliminating nuisance parameters. Statistical Science 14 (1):1–22.
   Web of Science ®Google Scholar
• Bernardo, J. M. 1979. Reference posterior distributions for Bayesian inference. Journal of the Royal Statistical Society, Series B 41:113–47 ( with discussion).
   Google Scholar
• Bernardo, J. M. 2005. Reference analysis. In Bayesian thinking: Modeling and computation, ed. D. K. Dey and C. R. Rao, 17–90. Amsterdam, The Netherlands: Elsevier.
   Google Scholar
• Bernardo, J. M., and J. M. Ramon. 1998. An introduction to Bayesian reference analysis: Inference on the ratio of multinomial parameters. Statistician 47:101–35.
   Google Scholar
• Bernardo, J. M., and A. F. M. Smith. 1994. Bayesian theory. New York, NY: Wiley.
   Google Scholar
• Berry, G., and P. Armitage. 1995. Mid-P confidence intervals: A brief review. Statistician 44 (4):417–23.
   Google Scholar
• Box, G. E. P., and G. C. Tiao. 1973. Bayesian inference in statistical analysis. New York, NY: Wiley Classics.
   Google Scholar
• Brown, L. D., T. Cai, and A. DasGupta. 2001. Interval estimation for a binomial proportion. Statistical Science 16 (2):101–33 ( with discussion).
   Web of Science ®Google Scholar
• Clopper, C. J., and E. S. Pearson. 1934. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26:404–16.
   Web of Science ®Google Scholar
• Davison, A. C. 2003. Statistical models. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Dawid, A. P., N. Stone, and J. V. Zidek. 1973. Marginalization paradoxes in Bayesian and structural inference. Journal of the Royal Statistical Society, Series B 35:189–233 ( with discussion).
   Google Scholar
• Draper, N. R., W. G. Hunter, and D. E. Tierney. 1969. Which product is better? Technometrics 11 (2):309–20.
   Web of Science ®Google Scholar
• Edwards, A. W. F. 1978. Commentary on the arguments of Thomas Bayes. Scandinavian Journal of Statistics 5:116–18.
   Web of Science ®Google Scholar
• Geisser, S. 1984. On prior distributions for binary trials. American Statistician 38 (4):244–51.
   Web of Science ®Google Scholar
• Geweke, J. 2005. Contemporary Bayesian econometrics and statistics. Hoboken, NJ: Wiley.
   Google Scholar
• Gottvall, K., C. Grunewald, and U. Waldenström. 2004. Safety of birth centre care: Perinatal mortality over a 10-year period. BJOG: An International Journal of Obstetrics and Gynaecology 111:71–78.
   PubMed Web of Science ®Google Scholar
• Gupta, R. C., R. A. Albanes, J. W. Penn, and T. J. White. 1997. Bayesian estimation of relative risk in biomedical research. Environmetrics 8:133–43.
   Web of Science ®Google Scholar
• Hartigan, J. A. 1983. Bayes theory. New York, NY: Springer-Verlag.
   Google Scholar
• Hashemi, L., B. Nandram, and R. Goldberg. 1997. Bayesian analysis for a single 2 × 2 table. Statistics in Medicine 16:1311–28.
   PubMed Web of Science ®Google Scholar
• Huzurbazar, V. S. 1976. Sufficient statistics. New York, NY: Marcel Dekker.
   Google Scholar
• Jeffreys, H. 1946. An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London—Series A 186 (1007):453–61.
   PubMed Web of Science ®Google Scholar
• Jeffreys, H. 1961. Theory of probability, 3rd ed., Oxford, UK: Oxford University Press.
   Google Scholar
• Jeffreys, H. 1980. Some general points in probability theory, In Bayesian analysis in econometrics and statistics, ed. A. Zellner, 451–54. Amsterdam, The Netherlands: North-Holland.
   Google Scholar
• Jovanovic, B. D., and P. S. Levy. 1997. A look at the rule of three. American Statistician 51 (2):137–39.
   Web of Science ®Google Scholar
• Lindley, D. V. 1965. Introduction to probability and statistics from a bayesian viewpoint. New York, NY: Cambridge University Press.
   Google Scholar
• Louis, T. A. 1981. Confidence intervals for a binomial parameter after observing no successes. American Statistician 35 (3):154.
   Web of Science ®Google Scholar
• Tebbs, J. M., R. B. Bilder, and B. K. Moser. 2001. An empirical Bayes group-testing approach to estimating small proportions. Communications in Statistics—Theory and Methods 32 (5):983–95.
   Web of Science ®Google Scholar
• Thomas, D. G., and J. J. Gart. 1977. A table of exact confidence limits for differences and ratios of two proportions and their odds ratios. Journal of the American Statistical Association 72 (357):73–76.
   Web of Science ®Google Scholar
• Tuyl, F., R. Gerlach, and K. Mengersen. 2008a. A comparison of Bayes–Laplace, Jeffreys, and other priors: the case of zero events. American Statistician 62 (1):40–44.
   Web of Science ®Google Scholar
• Tuyl, F., R. Gerlach, and K. Mengersen. 2008b. Inference for proportions in a 2 × 2 contingency table: HPD or not HPD? Biometrics 64 (4):1293–96.
   PubMed Web of Science ®Google Scholar
• Tuyl, F., R. Gerlach, and K. Mengersen. 2009a. Posterior predictive arguments in favor of the Bayes–Laplace prior as the consensus prior for binomial and multinomial parameters. Bayesian analysis 4 (1):151–58.
   Web of Science ®Google Scholar
• Tuyl, F., R. Gerlach, and K. Mengersen. 2009b. The rule of three, its variants and extensions. International Statistical Review 77 (2):266–75.
   Web of Science ®Google Scholar