Approximating hierarchical normal priors using a vague component

This article is concerned with hierarchical prior distributions and the effect of replacing the distribution of a component in the hierarchy with a diffuse distribution where all nondiffuse distributions are multivariate normal. Let f denote the posterior density function and g = gm, the approximation to f obtained by truncating the hierarchy at stage m. The Kullback-Leibler information index, I(f, g) = [integral operator] f log(f/g), will be used to measure the accuracy of g to avoid declaring specific objectives such as estimation or prediction. It is intuitively plausible that g will be increasingly more accurate as m increases; we show by theorems and two examples that this is sometimes but not always true. In the second example the behavior of I(f, g) depends on both the data and f's parameters and it may reach a minimum at an intermediate value of m which would then be optimal.