On resolving the Savage-Dickey paradox

TL;DR

This paper clarifies the measure-theoretic foundations of the Savage-Dickey ratio, showing it as a general Bayes factor representation and comparing it with other approximation methods.

Contribution

It reveals that the Savage-Dickey ratio is a general measure-theoretic representation of the Bayes factor and extends previous formulations with new comparisons.

Findings

Clarified the measure-theoretic basis of the Savage-Dickey ratio

Generalized Verdinelli and Wasserman's approach

Compared the new approximation with bridge sampling and Chib's methods

Abstract

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 the Savage-Dickey representation is in fact a generic representation of the Bayes factor that relies on specific measure-theoretic versions of the densities involved in the ratio, instead of a special identity imposing the above constraints on the prior distributions. We completely clarify the measure-theoretic foundations of the representation as well as the generalisation 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.