Extensions of the solidarity principle of the spectral gap for Gibbs samplers to their blocked and collapsed variants

Abstract: Connections of a spectral nature are formed between Gibbs samplers and their blocked and collapsed variants. The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps. This generalized solidarity principle is employed to establish that every cycle and mixture of Gibbs steps, which includes blocked Gibbs samplers and collapsed Gibbs samplers, inherits a spectral gap from a full Gibbs sampler. Exact relations between the spectra corresponding to blocked and collapsed variants of a Gibbs sampler are also established. An example is given to show that a blocked or collapsed Gibbs sampler does not in general inherit geometric ergodicity or a spectral gap from another blocked or collapsed Gibbs sampler.