Jim Hobert, University of Florida

Stability Relationships Among the Gibbs Sampler and its Subchains

Let $\Phi=\left\{(X_i,Y_i): i=0,1,2,\dots\right\}$ denote the Markov chain resulting from application of the two-variable Gibbs sampler using conditional densities that may correspond to an improper joint density. It is well known that the $X_i$'s themselves constitute a Markov chain as, of course, do the $Y_i$'s. We call these the subchains and denote them by $\Phi_x$ and $\Phi_y$. Our main result is that all three of these Markov chains share the same stability; that is, if one is positive recurrent (null recurrent, transient), then all three are positive recurrent (null recurrent, transient).

Our first application involves decision theory. Suppose that $W$ is a random variable with density $f(w|\theta)$ and that $\pi(\theta|w)$ is a proper posterior corresponding to an improper prior $\nu(\theta)$. Eaton (1992, Annals) showed that recurrence of the Markov chain with transition density $R(\eta|\theta)=\int \pi(\eta|w)f(w|\theta)dw$ implies that $\nu(\theta)$ is a "good" prior. We demonstrate that Eaton's Markov chain can be written as one of the subchains. Thus, recurrence of Eaton's chain can be established by showing that the other subchain is recurrent.

Our second application concerns Gibbs sampling with improper posteriors. Specifically, we show that even when the three chains $\Phi$, $\Phi_x$ and $\Phi_y$ are non-positive, there may still be positive recurrent chains lurking about. A recent example of Meng and van Dyk (1999, Biometrika) shows that it is possible to use these lurking chains to make valid statistical inferences via a Gibbs sampler based on an improper posterior. (Part of this talk is based on joint work with Christian Robert.)