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

Xavier Mak, James P. Hobert

TL;DR

The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps 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.

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.

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

TL;DR

The solidarity principle of the spectral gap for full Gibbs samplers is generalized to different cycles and mixtures of Gibbs steps 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.

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.
Paper Structure (16 sections, 19 theorems, 68 equations, 5 algorithms)

This paper contains 16 sections, 19 theorems, 68 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov operators
  4. Gibbs steps
  5. Cycles and mixtures of Gibbs steps
  6. A generalized solidarity principle of the spectral gap
  7. Inheritance of a spectral gap from a full Gibbs sampler
  8. Inheritance for cycles and mixtures of Gibbs steps
  9. Connections to collapsed Gibbs steps
  10. Commuting pairs of Gibbs steps
  11. Marginal chains of two-component deterministic-scan Gibbs samplers
  12. A spectral disconnection
  13. Background on geometric ergodicity
  14. Example
  15. Proof of Lemma \ref{['lem: P in P_sum has norm less than 1 for some choice of weights implies norm less than 1 for any choice of weights']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Every cycle and mixture of Gibbs steps, collapsed or otherwise, has a spectral gap whenever a full Gibbs sampler has a spectral gap.

Theorems & Definitions (36)

  • Theorem 1.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Remark 3.4
  • Corollary 3.5
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • ...and 26 more