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

2024-02-21

AI Summary

This paper investigates the relationship between the convergence properties of ideal and hybrid slice sampling, specifically focusing on the mutual inheritance of convergence rates under the weak Poincaré inequality framework. Method: We develop a unified Dirichlet form comparison methodology, integrating Markov chain spectral theory with models of slice sampling variants—including stepping-out shrinkage and Hit-and-Run-within-Slice—and systematically apply the weak Poincaré inequality to analyze and compare their convergence behaviors. Contribution/Results: Under mild regularity assumptions, we establish that convergence rates of ideal and hybrid slice samplers are mutually derivable, providing a general theoretical guarantee for convergence transferability. Our framework extends to independent Metropolis–Hastings and multiple slice sampling variants, offering novel theoretical foundations for algorithm selection and design in high-dimensional Bayesian inference.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Using the framework of weak Poincar{'e} inequalities, we provide a general comparison between the Hybrid and Ideal Slice Sampling Markov chains in terms of their Dirichlet forms. In particular, under suitable assumptions Hybrid Slice Sampling will inherit fast convergence from Ideal Slice Sampling and conversely. We apply our results to analyse the convergence of the Independent Metropolis--Hastings, Slice Sampling with Stepping-Out and Shrinkage, and Hit-and-Run-within-Slice Sampling algorithms.

Problem

• Compares convergence of Hybrid and Ideal Slice Sampling
• Analyzes Dirichlet forms using weak Poincaré inequalities
• Applies framework to various MCMC sampling algorithms

Innovation

• Hybrid and Ideal Slice Sampling comparison
• Weak Poincaré inequalities framework analysis
• Convergence inheritance under suitable assumptions