Open Access
2010 On resolving the Savage–Dickey paradox
Jean-Michel Marin, Christian P. Robert
Electron. J. Statist. 4: 643-654 (2010). DOI: 10.1214/10-EJS564


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.


Download Citation

Jean-Michel Marin. Christian P. Robert. "On resolving the Savage–Dickey paradox." Electron. J. Statist. 4 643 - 654, 2010.


Published: 2010
First available in Project Euclid: 9 July 2010

zbMATH: 1329.62091
MathSciNet: MR2660536
Digital Object Identifier: 10.1214/10-EJS564

Keywords: Bayes factor , Bayesian model choice , bridge sampling , conditional distribution , Hypothesis testing , Savage–Dickey ratio , zero measure set

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

Back to Top