Modifying Gibbs sampling to avoid self transitions (2403.18054v1)

Abstract: Gibbs sampling repeatedly samples from the conditional distribution of one variable, x_i, given other variables, either choosing i randomly, or updating sequentially using some systematic or random order. When x_i is discrete, a Gibbs sampling update may choose a new value that is the same as the old value. A theorem of Peskun indicates that, when i is chosen randomly, a reversible method that reduces the probability of such self transitions, while increasing the probabilities of transitioning to each of the other values, will decrease the asymptotic variance of estimates. This has inspired two modified Gibbs sampling methods, originally due to Frigessi, et al and to Liu, though these do not always reduce self transitions to the minimum possible. Methods that do reduce the probability of self transitions to the minimum, but do not satisfy the conditions of Peskun's theorem, have also been devised, by Suwa and Todo. I review past methods, and introduce a broader class of reversible methods, based on what I call "antithetic modification", which also reduce asymptotic variance compared to Gibbs sampling, even when not satisfying the conditions of Peskun's theorem. A modification of one method in this class reduces self transitions to the minimum possible, while still always reducing asymptotic variance compared to Gibbs sampling. I introduce another new class of non-reversible methods based on slice sampling that can also minimize self transition probabilities. I provide explicit, efficient implementations of all these methods, and compare their performance in simulations of a 2D Potts model, a Bayesian mixture model, and a belief network with unobserved variables. The non-reversibility produced by sequential updating can be beneficial, but no consistent benefit is seen from the individual updates being done by a non-reversible method.

• The paper introduces modified Gibbs sampling techniques that minimize self transitions through novel nested antithetic modifications.
• It leverages Peskun's theorem to theoretically guarantee reduced asymptotic variance compared to traditional Gibbs sampling.
• Empirical tests on models such as the Potts model and Bayesian mixtures confirm the methods' enhanced efficiency in MCMC simulations.

An Evaluation of Modified Gibbs Sampling Techniques to Minimize Self Transitions

The paper "Modifying Gibbs Sampling to Avoid Self Transitions" presents a significant contribution to the domain of improving efficiency in Markov Chain Monte Carlo (MCMC) sampling methods, specifically for Gibbs sampling. The focus of the paper, authored by Radford M. Neal, is on examining and introducing methods to reduce the self-transition probabilities in Gibbs sampling to enhance the statistical efficiency of the sampling process.

Overview

Gibbs sampling remains a fundamental MCMC technique due to its simplicity and direct applicability to a wide range of models within statistics, machine learning, and statistical physics. However, a critical inefficiency arises when the updated variable retains its previous value during a transition, labeled as a self-transition. This paper explores methods to minimize such self-transitions, which can reduce the asymptotic variance of estimators derived from sample paths, thus enhancing estimation efficiency.

Key Methods Examined

The paper investigates and builds upon pre-existing strategies like the Metropolis-Hastings Gibbs Sampling (MHGS) proposed by Liu, and the technique of Upward Nested Antithetic Modifications (UNAM), initially introduced by Frigessi et al., exploring their theoretical and practical advantages. These approaches leverage Peskun's theorem, which states that reducing self-transition probabilities, while maintaining detailed balance, generally improves estimator variance.

Further advancing beyond these, the author introduces a broader class, nested modifications, including Downward Nested Antithetic Modification (DNAM) and Zero-self DNAM (ZDNAM). These methods attempt to systematically reduce self-transitions to the theoretical minimum through an ordered adjustment of sampling probabilities, which Neal rigorously proves to efficiency-dominate standard Gibbs sampling under several conditions. ZDNAM particularly minimizes the probability of self-transitions to zero where feasible, offering improvements in scenarios where previous methods are constrained.

Numerical and Empirical Assessments

A comprehensive empirical analysis is conducted using simulations on the Potts model, Bayesian mixture models, and belief networks. These simulations demonstrate the conditions under which each methodology provides superior performance regarding asymptotic variance reduction across different update schemes (e.g., random, systematic). Notably, methods like ZDNAM not only minimize unnecessary self-transitions but also exhibit a strong reduction in variance compared to classical Gibbs sampling and even MHGS or UNAM under certain scan orders.

The experimental results suggest that non-reversible methods, despite theoretical challenges in their analysis, can outperform reversible methods. Sequential update schemes often perform better in practice than updates on randomly selected variables, providing a practical insight into MCMC efficiency improvements.

Implications and Future Research

This contribution has substantial theoretical and practical implications. By systematically minimizing self-transitions, these methods can serve as crucial components in more extensive MCMC frameworks, potentially improving computational efficiency in tasks like hierarchical model sampling and Bayesian inference.

Future research may explore optimizing these methods further, particularly by exploring hybrid strategies that incorporate insights from non-reversible and directional proposals, offering a new frontier in the exploration of sampling efficiency. Additionally, developing theory to address non-reversible strategies uniformly would further augment their applicability across diverse MCMC tasks.

In conclusion, Neal's work on modifying Gibbs Sampling elucidates a path toward more efficient statistical sampling methods by curbing the inefficiencies attributed to self-transitions. This avenue holds promise for substantial performance enhancements in both theoretical explorations and applied statistical modeling.