Geometric Ergodicity and Scanning Strategies for Two-Component Gibbs Samplers

In Markov chain Monte Carlo analysis, rapid convergence of the chain to its target distribution is crucial. A chain that converges geometrically quickly is geometrically ergodic. We explore geometric ergodicity for two-component Gibbs samplers (GS) that, under a chosen scanning strategy, evolve through one-at-a-time component-wise updates. We consider three such strategies: composition, random sequence, and random scans. We show that if any one of these scans produces a geometrically ergodic GS, so too do the others. Further, we provide a simple set of sufficient conditions for the geometric ergodicity of the GS. We illustrate our results using two examples.