  We consider versions of the Metropolis algorithm which avoid the inefficiency
of rejections. We first illustrate that a natural Uniform Selection Algorithm
might not converge to the correct distribution. We then analyse the use of
Markov jump chains which avoid successive repetitions of the same state. After
exploring the properties of jump chains, we show how they can exploit
parallelism in computer hardware to produce more efficient samples. We apply
our results to the Metropolis algorithm, to Parallel Tempering, to a Bayesian
model, to a two-dimensional ferromagnetic 4 x 4 Ising model, and to a
pseudo-marginal MCMC algorithm.
