An Encompassing Prior Generalization of the Savage-Dickey Density Ratio Test

Hoijtink, Klugkist, and colleagues proposed an encompassing prior (EP) approach to facilitate Bayesian model selection in nested models with inequality constraints. In this approach, samples are drawn from the prior and posterior distributions for an encompassing model that contains an inequality restricted version as a special case. The evidence in favor of the inequality restriction (i.e., the Bayes factor) simplifies to the ratio of the proportions of posterior and prior samples consistent with the inequality restriction. Up to now, this elegant formalism has been applied almost exclusively to models with inequality or “about equality” constraints. Here we show that the EP approach naturally extends to exact equality constraints by considering the ratio of the heights for the posterior and prior distributions at the point that is subject to test (i.e., the Savage-Dickey density ratio test). Therefore, the EP approach generalizes the Savage-Dickey test and can account for both inequality and equality constraints. The general EP approach is an elegant and computationally efficient procedure to calculate Bayes factors for nested models.