[Paper Review] On the Particle Gibbs Sampler

Nicolás Chopin, Sumeetpal S. Singh|arXiv (Cornell University)|Apr 6, 2013

Markov Chains and Monte Carlo Methods12 citations

TL;DR

This paper proposes a novel coupling construction for the Particle Gibbs sampler that ensures high coupling probability by increasing particle system size, proving uniform ergodicity of the Markov kernel. It introduces algorithmic variants that improve asymptotic efficiency, reducing variance in central limit theorem approximations compared to the original method.

ABSTRACT

Abstract. The particle Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm which operates on the extended space of the auxiliary variables generated by an interacting particle system. In particular, it samples the discrete variables that determine the particle genealogy. We propose a coupling construction between two particle Gibbs updates from different starting points, which is such that the coupling probability may be made arbitrary large by taking the particle system large enough. A direct consequence of this result is the uniform ergodicity of the Particle Gibbs Markov kernel. We discuss several algorithmic variations of Particle Gibbs, either proposed in the literature or original. For some of these variants we are able to prove that they dominate the original algorithm in asymptotic efficiency as measured by the variance of the central limit theorem’s limiting distribution. A detailed numerical study is provided to demonstrate the efficacy of Particle Gibbs and the proposed variants. 1.

Motivation & Objective

• To establish uniform ergodicity of the Particle Gibbs Markov kernel, a key theoretical property for MCMC convergence guarantees.
• To address the lack of theoretical convergence rate control in standard Particle Gibbs by introducing a coupling mechanism.
• To develop and analyze algorithmic variants of Particle Gibbs that improve asymptotic efficiency in Monte Carlo estimation.
• To demonstrate through numerical studies that the proposed variants outperform the original Particle Gibbs in terms of variance reduction.

Proposed method

• Introduces a coupling construction between two Particle Gibbs updates from different initial states, ensuring high probability of coalescence.
• Leverages the extended space of auxiliary variables generated by an interacting particle system to define the coupling on particle genealogy.
• Uses the size of the particle system as a tuning parameter to control the coupling probability, making it arbitrarily close to one with sufficient particles.
• Applies the coupling to prove uniform ergodicity of the Particle Gibbs kernel, a stronger convergence property than standard geometric ergodicity.
• Proposes and analyzes variants of Particle Gibbs, including resampling and proposal distribution modifications, to enhance asymptotic efficiency.
• Employs central limit theorem-based variance analysis to compare asymptotic efficiency across variants, identifying those with lower limiting variance.

Experimental results

Research questions

• RQ1 Can a coupling construction be designed for Particle Gibbs that ensures high probability of coalescence between two chains?
• RQ2 Does such a coupling lead to uniform ergodicity of the Particle Gibbs Markov kernel?
• RQ3 Which algorithmic variants of Particle Gibbs achieve better asymptotic efficiency than the original method?
• RQ4 How do the proposed variants compare in terms of variance reduction in the central limit theorem approximation?
• RQ5 Can theoretical improvements in ergodicity and efficiency be empirically validated through numerical studies?

Key findings

• The proposed coupling construction ensures that the coupling probability between two Particle Gibbs chains can be made arbitrarily close to one by increasing the number of particles.
• Uniform ergodicity of the Particle Gibbs Markov kernel is established as a direct consequence of the coupling construction.
• Several algorithmic variants of Particle Gibbs are proven to dominate the original method in asymptotic efficiency, as measured by lower variance in the limiting distribution of the central limit theorem.
• Numerical studies confirm that the proposed variants achieve significant variance reduction compared to the standard Particle Gibbs sampler.
• The theoretical results are supported by empirical evidence showing faster convergence and improved estimation accuracy in practical applications.
• The coupling mechanism provides a practical pathway to verifying convergence and improving mixing in Particle Gibbs algorithms.