Bayes factors for nonlinear hypotheses and likelihood distributions

Abstract

SUMMARY New methods are proposed which allow Bayes factors to be computed for hypotheses which impose nonlinear restrictions on the parameters. Projection methods are used to induce the prior distribution over the restricted parameter space which is required for computation of the Bayes factor. Various distance metrics are introduced to define the projection, including a utility-based metric which gives Kullback-Leibler divergence as a special case. Draws from the restricted and unrestricted prior distributions are used to construct marginal distributions of the likelihood which is shown to have additional diagnostic value over and above the Bayes factor. These methods are applied to hypotheses Statisticians are frequently asked to entertain hypotheses which are nonlinear in the model parameters. Nonlinear hypotheses arise both from subject matter theory as well as from the process of model selection. To address nonlinear hypotheses, researchers use frequentist methods which are usually implemented using asymptotic approximations. Berger (1987), Zellner (1971, 1978, 1984) and others have discussed the weaknesses of the classical significance testing approach and suggest Bayesian alternatives such as odds ratios. The odds ratio is the ratio of posterior probabilities of two competing hypotheses and is the product of the prior odds ratio and the Bayes factor. The odds ratio has the conceptual advantages of providing a natural probability metric for assessing the strength of data and prior beliefs. Because of the difficulties of assessing priors under the nonlinear restrictions and the computation of the required integrals, few Bayes factors have been computed for nonlinear hypotheses; see Lindley (1988) for an exception. In the present paper, we outline a general method for computing Bayes factors which is flexible, conceptually appealing and computationally tractable. The literature on Bayes factors starts with the work of Jeffreys (1961). Other references include Berger & Delampady (1987), Dickey (1971), Good (1985), Kass & Vaidyanathan (1992), Smith & Spiegelhalter (1980) and Zellner (1971, 1978, 1984). These papers deal with the testing of linear restrictions on the parameter space, often in the context of linear models. Even the widely-used Schwarz (1978) model selection criterion considers only linear restrictions in models of the exponential family form. The key difficulty which has kept researchers from developing Bayes factors for nonlinear hypotheses is the