## Bayesian Analysis

### Posterior predictive arguments in favor of the Bayes-Laplace prior as the consensus prior for binomial and multinomial parameters

#### Abstract

It is argued that the posterior predictive distribution for the binomial and multinomial distributions, when viewed via a hypergeometric-like representation, suggests the uniform prior on the parameters for these models. The argument is supported by studying variations on an example by Fisher, and complements Bayes' original argument for a uniform prior predictive distribution for the binomial. The fact that both arguments lead to invariance under transformation is also discussed.

#### Article information

Source
Bayesian Anal. Volume 4, Number 1 (2009), 151-158.

Dates
First available in Project Euclid: 22 June 2012

https://projecteuclid.org/euclid.ba/1340370393

Digital Object Identifier
doi:10.1214/09-BA405

Mathematical Reviews number (MathSciNet)
MR2486242

Zentralblatt MATH identifier
1330.62156

#### Citation

Tuyl, Frank; Gerlach, Richard; Mengersen, Kerrie. Posterior predictive arguments in favor of the Bayes-Laplace prior as the consensus prior for binomial and multinomial parameters. Bayesian Anal. 4 (2009), no. 1, 151--158. doi:10.1214/09-BA405. https://projecteuclid.org/euclid.ba/1340370393

#### References

• Agresti, A. and Min, Y. (2005). “Frequentist performance of Bayesian confidence intervals for comparing proportions in 2$\times$2 contingency tables.” Biometrics, 61: 515–523.
• Bayes, T. R. (1763). “An essay towards solving a problem in the doctrine of chances.” Phil. Trans. Roy. Soc. London, 53: 370–418.
• Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.
• Bernardo, J. M. (1979). “Reference posterior distributions for Bayesian inference.” Journal of the Royal Statistical Society, Series B, 41: 113–147 (with discussion).
• — (2005). “Reference analysis.” In Dey, D. K. and Rao, C. R. (eds.), Bayesian thinking: modeling and computation, 17–90. Amsterdam: Elsevier.
• Bernardo, J. M. and Ramon, J. M. (1998). “An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters.” The Statistician, 47: 101–135.
• Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. New York: Wiley.
• Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. New York: Wiley Classics.
• Edwards, A. W. F. (1978). “Commentary on the arguments of Thomas Bayes.” The Scandinavian Journal of Statistics, 5: 116–118.
• Fisher, R. A. (1973). Statistical Methods and Scientific Inference. New York: Hafner Press.
• Geisser, S. (1984). “On prior distributions for binary trials.” The American Statistician, 38(4): 244–251.
• — (1993). Predictive Inference: An Introduction. London: Chapman and Hall.
• Hashemi, L., Nandram, B., and Goldberg, R. (1997). “Bayesian analysis for a single 2$\times$2 table.” Statistics in Medicine, 16: 1311–1328.
• Huzurbazar, V. S. (1976). Sufficient Statistics. New York: Marcel Dekker.
• Jaynes, E. T. (2003). Probability Theory - The Logic of Science. Cambridge: University Press.
• Phillips, P. C. B. (1991). “To criticize the critics: an objective Bayesian analysis of stochastic trends.” Journal of Applied Econometrics, 6(4): 333–364.
• Stigler, S. M. (1982). “Thomas Bayes’s Bayesian inference.” Journal of the Royal Statistical Society, Series A, 145(2): 250–258.
• Thatcher, A. R. (1964). “Relationships between Bayesian and confidence limits for prediction.” Journal of the Royal Statistical Society, Series B, 26: 126–210.
• Welsh, A. H. (1996). Aspects of Statistical Inference. New York: John Wiley and Sons Ltd.