On resolving the Savage-Dickey paradox

The Savage-Dickey ratio is known as a specialised representation of the Bayes factor (O'Hagan and Forster, 2004) that allows for a functional plugging approximation of this quantity. We demonstrate here that it is a generic approximation method instead of an identity imposing constraints on the prior distributions, while incidentally clarifying the measure-theoretic bases of the method. We provide furthermore a general framework to produce a converging approximation of the Bayes factor that is unrelated with the earlier approach of Verdinelli and Wasserman (1995).