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Suppose that X is a random variable with density f(x|θ) and that π(θ|x) is a proper posterior corresponding to an improper prior ν(θ). The prior is called P-admissible if the generalized Bayes estimator of every bounded function of θ is almost-ν-admissible under squared error loss. Eaton showed that recurrence of the Markov chain with transition density R(η|θ) = ∫π(η|x)f(x|θ) dx is a sufficient condition for P-admissibility of ν(θ). We show that Eaton's Markov chain is recurrent if and only if its conjugate partner, with transition density R(y|x) = ∫f(y|θ)π(θ|x) dθ, is recurrent. This provides a new method of establishing P-admissibility. Often, one of these two Markov chains corresponds to a standard stochastic process for which there are known results on recurrence and transience. For example, when X is Poisson(θ) and an improper gamma prior is placed on θ, the Markov chain defined by R(y|x) is equivalent to a branching process with immigration. We use this type of argument to establish P-admissibility of some priors when f is a negative binomial mass function and when f is a gamma density with known shape.

Eaton's Markov chain, its conjugate partner and $\mathscr{P}$-admissibility