We consider admissibility in a formal Bayes setting. This includes the
frequentist concern of evaluating a decision rule and the Bayesian concern of
evaluating a prior. We generalize Eaton's method, which exploits a connection
between admissibility and a Markov chain defined by the sampling distribution
and posterior. This generalization leads us to introduce the idea of
$\varPhi$-admissibility, itself a generalization of strong admissibility. To
illustrate the method, we establish $\varPhi$-admissibility conditions for a
family of priors on multivariate normal means.