An encompassing prior generalization of the SavageDickey density ratio
An encompassing prior (EP) approach to facilitate Bayesian model selection for nested models with inequality constraints has been previously proposed. In this approach, samples are drawn from the prior and posterior distributions of an encompassing model that contains an inequality restricted version as a special case. The Bayes factor in favor of the inequality restriction then simplifies to the ratio of the proportions of posterior and prior samples consistent with the inequality restriction. This formalism has been applied almost exclusively to models with inequality or about equality constraints. It is shown 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 SavageDickey density ratio). The EP approach generalizes the SavageDickey ratio method, and can accommodate both inequality and exact equality constraints. The general EP approach is found to be a computationally efficient procedure to calculate Bayes factors for nested models. However, the EP approach to exact equality constraints is vulnerable to the BorelKolmogorov paradox, the consequences of which warrant careful consideration.
bayesian model selection inequality constraints equality constraints borelkolmogorov paradox hypothesis testing