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.Comment: Published in at http://dx.doi.org/10.1214/11-AAP806 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note:
substantial text overlap with arXiv:1001.279
