  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 $\nu(\|\theta\|^{2})d\theta$.
