Stability relationships among the Gibbs sampler and its subchains

The use of Gibbs samplers driven by improper posteriors has been a controve rsial issue in the statistics literature over the last few years. It has re cently been demonstrated that it is possible to make valid statistical infe rences through such Gibbs samplers. Furthermore, theoretical and empirical evidence has been given to support the idea that there are actually computa tional advantages to using these nonpositive recurrent Markov chains rather than more standard positive recurrent chains. These results provide motiva tion for a general study of the behavior of the Gibbs Markov chain when it is not positive recurrent. This article concerns stability relationships am ong the two-variable Gibbs sampler and its subchains. We show that these th ree Markov chains always share the same stability; that is, they are either all positive recurrent, all null recurrent, or all transient. In addition, we establish general results concerning the ways in which positive recurre nt Markov chains can arise from null recurrent and transient Gibbs chains. Six examples of varying complexity are used to illustrate the results.