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

TL;DR

This paper generalizes the solidarity principle of the spectral gap for Gibbs samplers to their blocked and collapsed variants, showing inheritance of spectral gaps under certain conditions and providing spectral relations between variants.

Contribution

It extends the spectral gap solidarity principle to cycles and mixtures of Gibbs steps, including blocked and collapsed versions, and establishes spectral relations between them.

Findings

Blocked and collapsed Gibbs samplers can inherit spectral gaps from full Gibbs samplers.

Exact spectral relations are derived for blocked and collapsed variants.

Inheritance of geometric ergodicity is not guaranteed between different blocked or collapsed samplers.

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.