  We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs
samplers, which update their selection probabilities (and perhaps also their
proposal distributions) on the fly during a run by learning as they go in an
attempt to optimize the algorithm. We present a cautionary example of how even
a simple-seeming adaptive Gibbs sampler may fail to converge. We then present
various positive results guaranteeing convergence of adaptive Gibbs samplers
under certain conditions.
