A Generalized Savage-Dickey Ratio

TL;DR

This paper introduces a generalized form of the Savage-Dickey Ratio using Radon-Nikodym derivatives, broadening its applicability to various probability spaces beyond those with Lebesgue densities.

Contribution

It extends the Savage-Dickey Ratio to a measure-theoretic framework, enabling its use in more general probabilistic models.

Findings

The generalized ratio is derived using measure theory.

Application to a distributional modeling problem demonstrates its practical utility.

Broader applicability to nested models without Lebesgue density assumptions.

Abstract

In this brief research note I present a generalized version of the Savage-Dickey Density Ratio for representation of the Bayes factor (or marginal likelihood ratio) of nested statistical models; the new version takes the form of a Radon-Nikodym derivative and is thus applicable to a wider family of probability spaces than the original (restricted to those admitting an ordinary Lebesgue density). A derivation is given following the measure-theoretic construction of Marin & Robert (2010), and the equivalent estimator is demonstrated in application to a distributional modeling problem.