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

Abstract

Suppose that X is a random variable with density $f(x|\theta)$ and that $\pi(\theta|x)$ is a proper posterior corresponding to an improper prior $\nu(\theta)$. The prior is called $\mathscr{P}$-admissible if the generalized Bayes estimator of every bounded function of $\theta$ is almost-$\nu$-admissible under squared error loss. Eaton showed that recurrence of the Markov chain with transition density $R(\eta|\theta) = \int \pi(\eta|x)f(x|\theta) dx$ is a sufficient condition for $\mathscr{P}$-admissibility of $\nu(\theta)$. We show that Eaton’s Markov chain is recurrent if and only if its conjugate partner, with transition density $\tilde{R}(y|x) = \int f(y|\theta) \pi(\theta|x) d\theta$, is recurrent. This provides a new method of establishing $\mathscr{P}$-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 $(\theta)$ and an improper gamma prior is placed on $\theta$, the Markov chain defined by $\tilde{R}(y|x)$ is equivalent to a branching process with immigration. We use this type of argument to establish $\mathscr{P}$-admissibility of some priors when $f$ is a negative binomial mass function and when $f$ is a gamma density with known shape.