Eaton’s Markov Chain, its Conjugate Partner and P-admissibility

Suppose that X is a random variable with density f(xj ) and that ( jx) is a proper posterior corresponding to an improper prior ( ). The prior is called P-admissible if the generalized Bayes estimator of every bounded function of is almost-admissible under squared error loss. Eaton (1992) showed that recurrence of the Markov chain with transition density R( j ) = R ( jx)f(xj )dx is a su cient condition for P-admissibility of ( ). We show that Eaton's Markov chain is recurrent if and only if its conjugate partner, with transition density ~ R(yjx) = R f(yj ) ( jx)d , 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( ) and an improper gamma prior is placed on , the Markov chain de ned by ~ R(yjx) 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.