Evaluating Default Priors with a Generalization of Eaton's Markov Chain

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $\Phi$ be a class of functions on the parameter space and consider estimating elements of $\Phi$ under quadratic loss. If the formal Bayes estimator of every function in $\Phi$ is admissible, then the prior is strongly admissible with respect to $\Phi$. Eaton's method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with the inferential setting. In previous work, this was handled differently depending upon whether $\phi \in \Phi$ was bounded or unbounded. We introduce and study a new Markov chain which allows us to unify and generalize existing approaches while simultaneously broadening the scope of their potential applicability. To illustrate the method, we establish strong admissibility conditions when the model is a $p$-dimensional multivariate normal distribution with unknown mean vector $\theta$ and the prior is of the form $\nu(\|\theta\|^{2})d\theta$.