Eaton's Markov chain, its conjugate partner and P-admissibility

Suppose that X is a random variable with density f(x\theta) and that pi(the ta\x) is a proper posterior corresponding to an improper prior nu(theta). T he prior is called P-admissible if the generalized Bayes estimator of every bounded function of theta is almost-nu-admissible under squared error loss . Eaten showed that recurrence of the Markov chain with transition density R(eta\theta) = integral pi(eta\x)f(x\theta) dx is a sufficient condition fo r P-admissibility of nu(theta). We show that Eaten's Markov chain is recurr ent if and only if its conjugate partner, with transition density (R) over tilde(y\x) = integral f(y\theta) pi(theta\x)d theta, is recurrent. This pro vides 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 Poi sson(theta) and an improper gamma prior is placed on theta, the Markov chai n defined by (R) over tilde(y\x) is equivalent to a branching process with immigration. We use this type of argument to establish P-admissibility of s ome priors when f is a negative binomial mass function and when f is a gamm a density with known shape.