Electronic Journal of Statistics

On resolving the Savage–Dickey paradox

Jean-Michel Marin and Christian P. Robert

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When testing a null hypothesis H0: θ=θ0 in a Bayesian framework, the Savage–Dickey ratio (Dickey, 1971) is known as a specific representation of the Bayes factor (O’Hagan and Forster, 2004) that only uses the posterior distribution under the alternative hypothesis at θ0, thus allowing for a plug-in version of this quantity. We demonstrate here that the Savage–Dickey representation is in fact a generic representation of the Bayes factor and that it fundamentally relies on specific measure-theoretic versions of the densities involved in the ratio, instead of being a special identity imposing some mathematically void constraints on the prior distributions. We completely clarify the measure-theoretic foundations of the Savage–Dickey representation as well as of the later generalisation of Verdinelli and Wasserman (1995). We provide furthermore a general framework that produces a converging approximation of the Bayes factor that is unrelated with the approach of Verdinelli and Wasserman (1995) and propose a comparison of this new approximation with their version, as well as with bridge sampling and Chib’s approaches.

Article information

Electron. J. Statist., Volume 4 (2010), 643-654.

First available in Project Euclid: 9 July 2010

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Zentralblatt MATH identifier

Bayesian model choice Bayes factor bridge sampling conditional distribution hypothesis testing Savage–Dickey ratio zero measure set


Marin, Jean-Michel; Robert, Christian P. On resolving the Savage–Dickey paradox. Electron. J. Statist. 4 (2010), 643--654. doi:10.1214/10-EJS564. https://projecteuclid.org/euclid.ejs/1278682959

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