Spectral Telescope: Convergence Rate Bounds for Random-Scan Gibbs Samplers Based on a Hierarchical Structure

08/24/2022
by   Qian Qin, et al.
0

Random-scan Gibbs samplers possess a natural hierarchical structure. The structure connects Gibbs samplers targeting higher dimensional distributions to those targeting lower dimensional ones. This leads to a quasi-telescoping property of their spectral gaps. Based on this property, we derive three new bounds on the spectral gaps and convergence rates of Gibbs samplers on general domains. The three bounds relate a chain's spectral gap to, respectively, the correlation structure of the target distribution, a class of random walk chains, and a collection of influence matrices. Notably, one of our results generalizes the technique of spectral independence, which has received considerable attention for its success on finite domains, to general state spaces. We illustrate our methods through a sampler targeting the uniform distribution on a corner of an n-cube.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/04/2023

Solidarity of Gibbs Samplers: the spectral gap

Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used i...
research
06/26/2020

Convergence Rates of Two-Component MCMC Samplers

Component-wise MCMC algorithms, including Gibbs and conditional Metropol...
research
01/28/2018

Adapting The Gibbs Sampler

The popularity of Adaptive MCMC has been fueled on the one hand by its s...
research
03/10/2021

Dimension free convergence rates for Gibbs samplers for Bayesian linear mixed models

The emergence of big data has led to a growing interest in so-called con...
research
04/29/2020

On the convergence complexity of Gibbs samplers for a family of simple Bayesian random effects models

The emergence of big data has led to so-called convergence complexity an...
research
10/29/2020

Rates of convergence for Gibbs sampling in the analysis of almost exchangeable data

Motivated by de Finetti's representation theorem for partially exchangea...
research
04/27/2021

Approximating the Spectral Gap of the Pólya-Gamma Gibbs Sampler

The self-adjoint, positive Markov operator defined by the Pólya-Gamma Gi...

Please sign up or login with your details

Forgot password? Click here to reset