Convergence Rates of Two-Component MCMC Samplers

Qian Qin; Galin L. Jones

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arXiv · 2006.14801

Convergence Rates of Two-Component MCMC Samplers

Qian Qin·Galin L. Jones

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Abstract

Component-wise MCMC algorithms, including Gibbs and conditional Metropolis-Hastings samplers, are commonly used for sampling from multivariate probability distributions. A long-standing question regarding Gibbs algorithms is whether a deterministic-scan (systematic-scan) sampler converges faster than its random-scan counterpart. We answer this question when the samplers involve two components by establishing an exact quantitative relationship between the $L^2$ convergence rates of the two samplers. The relationship shows that the deterministic-scan sampler converges faster. We also establish qualitative relations among the convergence rates of two-component Gibbs samplers and some conditional Metropolis-Hastings variants. For instance, it is shown that if some two-component conditional Metropolis-Hastings samplers are geometrically ergodic, then so are the associated Gibbs samplers.

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Qian Qin, Galin L. Jones. 2021-05-09. Convergence Rates of Two-Component MCMC Samplers. https://arxiv.org/abs/2006.14801

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