Adaptive Gibbs samplers and related MCMC methods

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 y during a run, by learning as they go in an attempt to optimise 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