A prior for the variance in hierarchical models

The choice of prior distributions for the variances can be important and quite difficult in Bayesian hierarchical and variance component models. For situations where little prior information is available, a 'noninformative' type prior is usually chosen. 'Noninformative' priors have been discussed by many authors and used in many contexts. However, care must be taken using these prior distributions as many are improper and thus, can lead to improper posterior distributions. Additionally, in small samples, these priors can be 'informative'. In this paper, we investigate a proper 'vague' prior, the uniform shrinkage prior (Strawderman 1971; Christiansen & Morris 1997). We discuss its properties and show how posterior distributions for common hierarchical models using this prior lead to proper posterior distributions. We also illustrate the attractive frequentist properties of this prior for a normal hierarchical model including testing and estimation. To conclude, we generalize this prior to the multivariate situation of a covariance matrix.