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On resolving the Savage-Dickey paradox. (English) Zbl 1329.62091

Summary: When testing a null hypothesis \(H_{0}: \theta =\theta _{0}\) in a Bayesian framework, the Savage-Dickey ratio [J. M. Dickey, Ann. Math. Stat. 42, 204–223 (1971; Zbl 0274.62020)] is known as a specific representation of the Bayes factor [A. O’Hagan and J. Forster, Kendall’s advanced theory of statistics. Volume 2B: Bayesian inference. 2nd edition. London: Arnold (2004; Zbl 1058.62002)] that only uses the posterior distribution under the alternative hypothesis at \(\theta _{0}\), 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 I. Verdinelli and L. Wasserman [J. Am. Stat. Assoc. 90, No. 430, 614–618 (1995; Zbl 0826.62022)]. 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 [loc. cit.] and propose a comparison of this new approximation with their version, as well as with bridge sampling and Chib’s approaches.

MSC:

62F03 Parametric hypothesis testing
62F15 Bayesian inference

Software:

R
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References:

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