Weak Poincar\'e inequality comparisons for ideal and hybrid slice sampling

TL;DR

This paper compares the convergence properties of Hybrid and Ideal Slice Sampling using weak Poincaré inequalities, showing conditions under which they inherit each other's efficiency, with applications to various Markov chain algorithms.

Contribution

It introduces a framework for comparing Hybrid and Ideal Slice Sampling via weak Poincaré inequalities, revealing their convergence relationships and applicability to multiple algorithms.

Findings

Hybrid inherits fast convergence from Ideal Slice Sampling under certain conditions.

The analysis applies to Independent Metropolis-Hastings, stepping-out, shrinkage, and Hit-and-Run algorithms.

The framework provides a unified way to compare sampling methods based on Dirichlet forms.

Abstract

Using the framework of weak Poincar\'e inequalities, we provide a general comparison between Hybrid and Ideal Slice Sampling in terms of their corresponding Dirichlet forms. In particular, under suitable assumptions Hybrid Slice Sampling inherits fast convergence from Ideal Slice Sampling and conversely. We apply our results to analyse the convergence of the Independent Metropolis-Hastings, stepping-out and shrinkage, as well as Hit-and-Run within slice sampling algorithms.