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The Savage-Dickey density ratio is a specific expression of the Bayes factorwhen testing a precise (equality constrained) hypothesis against anunrestricted alternative. The expression greatly simplifies the computation ofthe Bayes factor at the cost of assuming a specific form of the prior under theprecise hypothesis as a function of the unrestricted prior. A generalizationwas proposed by Verdinelli and Wasserman (1995) such that the priors can befreely specified under both hypotheses while keeping the computationaladvantage. This paper presents an extension of this generalization when thehypothesis has equality as well as order constraints on the parameters ofinterest. The methodology is used for a constrained multivariate t test usingthe JZS Bayes factor and a constrained hypothesis test under the multinomialmodel.

A Generalization of the Savage-Dickey Density Ratio for Testing Equality and Order Constrained Hypotheses