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Abstract:
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We consider situations in Bayesian analysis where the prior is indexed by a hyperparameter taking on a continuum of values. We distinguish some arbitrary value of the hyperparameter, and consider the problem of estimating the Bayes factor for the model indexed by the hyperparameter vs. the model specified by the distinguished point, as the hyperparameter varies. We assume that we have Markov chains from the posterior for a few of the priors, and develop a method for efficiently computing estimates of the entire family of Bayes factors. As an application, we show how our methodology can be used to select the hyperparameters in a standard model for variable selection in Bayesian linear regression, in which a hierarchical prior involves a prior distribution on which variables go into the model and then a prior distribution on the regression parameters for the selected variables.
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