Weak Poincaré inequality comparisons for ideal and hybrid slice sampling

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.

• The paper introduces a framework that uses weak Poincaré inequalities to compare the convergence behavior of ideal and hybrid slice sampling methods.
• It rigorously derives Dirichlet form formulations and conditions under which hybrid techniques nearly match the efficiency of ideal sampling.
• The study provides performance bounds and insights that enhance algorithm selection in computational Bayesian inference and classical Metropolis methods.

Weak Poincaré Inequality Comparisons for Ideal and Hybrid Slice Sampling

The paper by Power, Rudolf, Sprungk, and Wang presents a methodological advance in the comparison of Ideal and Hybrid Slice Sampling methods via weak Poincaré inequalities. This framework scrutinizes the convergence properties of these Markov chains through the evaluation of their Dirichlet forms. The primary achievement lies in establishing that Hybrid Slice Sampling retains fast convergence properties from its ideal counterpart, assuming suitable conditions are met.

Key Contributions

1. Framework Establishment: The paper formalizes a framework to compare the convergence of Markov chains using weak Poincaré inequalities. Through this framework, the authors provide conditions under which the Hybrid Slice Sampling can mirror the convergence properties of the Ideal Slice Sampling.
2. Algorithm Analysis: By applying the framework, the paper evaluates several algorithms, including Independent Metropolis-Hastings and methods utilizing Slice Sampling with Stepping-Out, Shrinkage, and Hit-and-Run. This is used to gauge the hybrid algorithms' efficiency in converging towards their target distributions compared to their ideal versions.
3. Mathematical Descriptions: The authors rigorously detail the mathematical formulations underlying the Dirichlet forms and weak Poincaré inequalities that form the backbone of their comparative analysis.

Methodology

• Dirichlet Form Representation: The study computes Dirichlet forms for both Ideal and Hybrid Slice Samplers and demonstrates how these can provide insights into the variance dissipation rates of the chains.
• Comparison of Convergence: The utilitarian aspect of weak Poincaré inequalities lies in quantifying how well a less feasible algorithm (Hybrid) mimics its theoretically more efficient counterpart (Ideal). They derive conditions under which one can establish that the hybrid methods, though easier to implement, are 'almost as efficient' in theoretical convergence.
• Algorithmic Implementation: The paper addresses practical challenges in implementing Ideal Slice Samplers due to the difficulty in sampling from complex distribution slices. Hybrid versions overcome this by using stochastic steps within slices using Markov kernels.

Numerical Results and Implications

• Performance Bounds: The authors provide explicit bounds on the convergence rates of various Hybrid Slice Sampling algorithms, relative to their Ideal counterparts, through a detailed mathematical treatment of weak Poincaré inequalities.
• Extensions to Classical Algorithms: Furthermore, the paper recasts classical Metropolis algorithms as instances of Hybrid Slice Sampling, thus providing a new perspective on their convergence properties. This links the efficiency of Hybrid Slice Sampling to well-known algorithms in computational statistics and facilitates comparison using established metrics.
• Implications for Computational Bayesian Inference: This analysis serves as a critical tool in Bayesian computation, where efficient exploration of state spaces is vital. Given the practical limitations and tuning requirements of classical Metropolis methods, the insights derived here could inform choices in algorithm selection and parameterization.

Future Directions

The proposed framework prompts further investigations into other Markov chains that may benefit from similar analyses using weak Poincaré inequalities. Additionally, it sets a foundation for considering additional modifications to Slice Sampling that could leverage the strengths of hybrid methods while minimizing their drawbacks. Advanced theoretical work could explore loosening assumptions to broaden the applicability of these comparisons or developing automatic tuning methods based on the insights provided.

Overall, this work advances the field by bridging theoretical constructs with practical algorithm implementations, facilitating enhanced sampling methods in computational statistics.