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Neutral noninformative and informative conjugate beta and gamma prior distributions. (English) Zbl 1271.62045
Summary: The conjugate binomial and Poisson models are commonly used for estimating proportions or rates. However, it is not well known that the conventional noninformative conjugate priors tend to shrink the posterior quantiles toward the boundary or toward the middle of the parameter space, making them thus appear excessively informative. The shrinkage is always largest when the number of observed events is small. This behavior persists for all sample sizes and exposures. The effect of the prior is therefore most conspicuous and potentially controversial when analyzing rare events. As alternative default conjugate priors, I introduce Beta(1/3, 1/3) and Gamma(1/3, 0), which I call ‘neutral’ priors because they lead to posterior distributions with approximately 50 per cent probability that the true value is either smaller or larger than the maximum likelihood estimate. This holds for all sample sizes and exposures as long as the point estimate is not at the boundary of the parameter space. I also discuss the construction of informative prior distributions. Under the suggested formulation, the posterior median coincides approximately with the weighted average of the prior median and the sample mean, yielding priors that perform more intuitively than those obtained by matching moments and quantiles.

MSC:
62F15 Bayesian inference
62E15 Exact distribution theory in statistics
Software:
BayesDA
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References:
[1] Agresti, A. and Coull, B. A. (1998). Approximate Is Better than “Exact” for Interval Estimation of Binomial Proportions., The American Statistician 52 119-126.
[2] Agresti, A. and Min, Y. (2001). On Small-Sample Confidence Intervals for Parameters in Discrete Distributions., Biometrics 57 963-971. · Zbl 1209.62041
[3] Bayarri, M. J. and Berger, J. O. (2004). The Interplay of Bayesian and Frequentist Analysis., Statistical Science 19 pp. 58-80. · Zbl 1062.62001
[4] Berg, C. and Pedersen, H. L. (2006). The Chen-Rubin Conjecture in a Continuous Setting., Methods and Applications of Analysis 13 63-88. · Zbl 1107.60006
[5] Berger, J. (2006). The case for objective Bayesian analysis., Bayesian Analysis 1 385-402. · Zbl 1331.62042
[6] Bernardo, J. M. (1979). Reference Posterior Distributions for Bayesian Inference., Journal of the Royal Statistical Society. Series B (Methodological) 41 113-147. · Zbl 0428.62004
[7] Bernardo, J. M. (2005). Reference Analysis. In, Handbook of Statistics 25: Bayesian Thinking, Modeling and Computation (D. K. Dey and C. R. Rao, eds.) 17-90. Elsevier, Amsterdam. · Zbl 0682.62018
[8] Box, G. E. P. and Tiao, G. C. (1973)., Bayesian Inference in Statistical Data Analysis , 1st ed. Wiley-Interscience, New York. · Zbl 0271.62044
[9] Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2004)., Bayesian Data Analysis , 2nd ed. Chapman & Hall/CRC, London. · Zbl 1039.62018
[10] Gelman, A., Jakulin, A., Pittau, M. G. and Yu, S.-S. (2008). A weakly informative default prior distrbution for logistic and other regression models., The Annals of Applied Statistics 2 1360-1383. · Zbl 1156.62017
[11] Haldane, J. B. S. (1948). The Precision of Observed Values of Small Frequencies., Biometrika 35 297-300. · Zbl 0039.36205
[12] Hanley, J. A. and Lippman-Hand, A. (1983). If nothing goes wrong, is everything all right? Interpreting zero numerators., Journal of the American Medical Association 249 1743-1745.
[13] Jeffreys, H. (1961)., Theory of probability , 3rd ed. Oxford University Press, New York. · Zbl 0116.34904
[14] Jovanovic, B. D. and Levy, P. S. (1997). A Look at the Rule of Three., The American Statistician 51 137-139.
[15] Kerman, J. (2011). A closed-form approximation for the median of the beta distribution. · Zbl 1271.62045
[16] Neuenschwander, B., Rouyrre, N., Hollaender, N., Zuber, E. and Branson, M. (2011). A proof of concept phase II non-inferiority criterion., Statistics in Medicine 30 1618-1627.
[17] O’Hagan, A., Buck, C. E., Daneshkhah, A., Eiser, J. R., Garthwaite, P. H., Jenkinson, D. J., Oakley, J. E. and Rakow, T. (2006)., Uncertain judgements: Eliciting experts’ Probabilities . Wiley, Hoboken, NJ. · Zbl 1269.62009
[18] Rubin, D. B. (1984). Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician., The Annals of Statistics 12 pp. 1151-1172. · Zbl 0555.62010
[19] Spiegelhalter, D. J., Abrams, K. R. and Myles, J. P. (2004)., Bayesian Approaches to Clinical Trials and Health-Care Evaluation . Wiley, Chichester. · Zbl 1057.62105
[20] Tuyl, F., Gerlach, R. and Mengersen, K. (2008). A comparison of Bayes-Laplace, Jeffreys, and other priors: the case of zero events., The American Statistician 62 40-44. · Zbl 05680768
[21] Winkler, R. L., Smith, J. E. and Fryback, D. G. (2002). The Role of Informative Priors in Zero-Numerator Problems: Being Conservative versus Being Candid., The American Statistician 56 1-4.
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