Comment on "Bayesian Computation and Stochastic Systems"

We congratulate the authors on a magnificent paper, providing a nicely paced introduction to Markov chain Monte Carlo and its applications, together with several new ideas. In particular the class of pairwise difference priors is bound to have a substantial impact on future applied work. Other ideas given less prominence in the paper are also valuable, for example, the construction of simultaneous credible regions based on MCMC output. There are several issues which we wish to comment on in detail