How many iterations in the Gibbs sampler

This technical report consists of three short papers on Monte Carlo Markov chain inference. The first paper, "How many iterations in the Gibbs sampkcr?, " propees an easily implemented method for determining the total number of iterations required to estimate probabilities and quantiles of the posterior distribution, and also the number of initial iterations that should be discarded to allow for "burn-in". The second paper discusses model determination via predictive distributions. The paper advocates the standard Bayesian procedure that uses Bayes factors, and points out that this can be implemented quite easily using sampling-based methods. The third paper discusses issues in spatial statistics that use sampling-based meth-ods. Several issues in the Bayesian image restoration approach are discussed: the modeling of spatial dependence, allowing for model uncertainty, the improper poste-rior distributions that arise in hierarcEcal Bayes modeling, and the modeling of local dependence between counts when it cannot be assumed that the observations are in-dependent given the true rates