Decomposing posterior variance

We propose a decomposition of posterior variance somewhat in the spirit of an ANOVA decomposition. Terms in this decomposition come in pairs. Given a single parametric model, for instance, one term describes uncertainty arising because the parameter value is unknown while the other describes uncertainty propagated via uncertainty about which prior distribution is appropriate for the parameter. In the context of multiple candidate models and model-averaged estimates, two additional terms emerge resulting in a four-term decomposition. In the context of multiple spaces of models, six terms result. The value of the decomposition is twofold. First, it yields a fuller accounting of uncertainty than methods which condition on data-driven choices of models or model spaces. Second, it constitutes a novel approach to the study of prior influence in Bayesian analysis.