Stephen Brooks and Andrew Gelman

We generalise the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously. Keywords: Markov chain Monte Carlo, Convergence Diagnosis, Inference y Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, England and Department of Statistics, Columbia University, New York, NY 10027, USA. We thank Gareth Roberts, Xiao-Li Meng, Phillip Price, three anonymous referees and an Associat..