This paper investigates a particular type of slice sampler algorithm, the polar slice sampler. This algorithm is shown to have convergence properties which are essentially independent of the dimension of the problem, at least for log-concave densities. For such densities, the algorithm provably converges (from appropriate starting point) to within 0:01 of stationarity in total variation distance in a number of iterations given as a computable function of the spherical asymmetry of the density. In particular, for spherically symmetric log-concave densities, in arbitrary dimension, with appropriate starting point, we prove that the algorithm converges in at most 525 iterations. Simulations are done which connrm the polar slice sampler's excellent performance.

Convergence diagnostics are widely used to determine how many initial “burn-in” iterations should be discarded from the output of a Markov chain Monte Carlo (MCMC) sampler in the hope that the remaining samples are representative of the target distribution of interest. This paper demonstrates that some ways of applying convergence diagnostics may actually introduce bias into estimation based on the sampler output. To avoid this possibility, we recommend choosing the number of burn-in iterations r by applying convergence diagnostics to one or more pilot chains, and then basing estimation and inference on a separate long chain from which the first r iterations have been discarded.

This paper introduces a way of constructing non-informative priors for Bayesian analysis, by taking a limit of priors arising from hierarchical constructions as the number of levels in the hierarchy converges to 1. Results are proved showing that for location families, and other related cases, limits are often not dependent on the exact form of the incremental distribution used.

We develop quantitative bounds on rates of convergence for continuous-time Markov processes on general state spaces. Our methods involve coupling and shift-coupling, and make use of minorization and drift conditions. In particular, we use auxiliary coupling to establish the existence of small (or pseudo-small) sets. We apply our method to some diiusion examples. We are motivated by interest in the use of Langevin diiusions for Monte Carlo simulation.

Richard Tweedie for helpful conversations. We thank the anonymous referee for a very helpful report, including simplifying the proof of Theorem 9 and pointing out Lemma 3 (c) herein.

Acknowledgements. We thank Torgny Lindvall for suggesting the use of shift-coupling, thank David Aldous, John Baxter, and Richard Tweedie for insightful conversations, and thank Alan Gelfand for organizing the conference at Mt. Holyoke College which initiated this collaboration.

We develop a theory of suuciency for Markov chains. In particular, we prove that to analyse convergence in total variation distance, it suuces to obtain bounds on a suucient chain.