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2011 Neutral noninformative and informative conjugate beta and gamma prior distributions
Jouni Kerman
Electron. J. Statist. 5: 1450-1470 (2011). DOI: 10.1214/11-EJS648


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.


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Jouni Kerman. "Neutral noninformative and informative conjugate beta and gamma prior distributions." Electron. J. Statist. 5 1450 - 1470, 2011.


Published: 2011
First available in Project Euclid: 4 November 2011

zbMATH: 1271.62045
MathSciNet: MR2851686
Digital Object Identifier: 10.1214/11-EJS648

Keywords: Bayesian inference , Beta distribution , conjugate analysis , gamma distribution , noninformative distributions , Prior distributions

Rights: Copyright © 2011 The Institute of Mathematical Statistics and the Bernoulli Society

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