Theoretical work on Markov chain Monte Carlo (MCMC) algorithms has
so far mainly concentrated on the properties of simple algorithms, such as the
Gibbs sampler, or the full-dimensional Hastings-Metropolis algorithm. In
practice, these simple algorithms are used as building blocks for more
sophisticated methods, which we shall refer to as hybrid samplers. It is
often hoped that good convergence properties (e.g., geometric ergodicity,
etc.) of the building blocks will imply similar properties of the hybrid
chains. However, little is rigorously known.
¶ In this paper, we concentrate on two special cases of hybrid
samplers. In the first case, we provide a quantitative result for the rate of
convergence of the resulting hybrid chain. In the second case, concerning the
combination of various Metropolis algorithms, we establish geometric
ergodicity.