On resolving the Savage–Dickey paradox

Abstract

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.