  Consider a parametric statistical model $P(\mathrm{d}x|\theta)$ and an
improper prior distribution $\nu(\mathrm{d}\theta)$ that together yield a
(proper) formal posterior distribution $Q(\mathrm{d}\theta|x)$. The prior is
called strongly admissible if the generalized Bayes estimator of every bounded
function of $\theta$ is admissible under squared error loss. Eaton [Ann.
Statist. 20 (1992) 1147--1179] has shown that a sufficient condition for strong
admissibility of $\nu$ is the local recurrence of the Markov chain whose
transition function is $R(\theta,\mathrm{d}\eta)=\int
Q(\mathrm{d}\eta|x)P(\mathrm {d}x|\theta)$. Applications of this result and its
extensions are often greatly simplified when the Markov chain associated with
$R$ is irreducible. However, establishing irreducibility can be difficult. In
this paper, we provide a characterization of irreducibility for general state
space Markov chains and use this characterization to develop an easily checked,
necessary and sufficient condition for irreducibility of Eaton's Markov chain.
All that is required to check this condition is a simple examination of $P$ and
$\nu$. Application of the main result is illustrated using two examples.
