Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles

Christophe Andrieu, Anthony W L Lee*, Sam Power, Andi Wang

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

1 Citation (Scopus)
1 Downloads (Pure)

Abstract

We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on R^d for any value of the proposal variance, which when scaled appropriately recovers the correct d^{−1} dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the L^2-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.
Original languageEnglish
Pages (from-to)4022-4071
Number of pages50
JournalAnnals of Applied Probability
Volume34
Issue number4
DOIs
Publication statusPublished - 1 Aug 2024

Bibliographical note

Publisher Copyright:
© 2024 Institute of Mathematical Statistics. All rights reserved.

Fingerprint

Dive into the research topics of 'Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles'. Together they form a unique fingerprint.

Cite this