Hybrid Gibbs samplers represent a prominent class of approximated Gibbs algorithms that utilize Markov chains to approximate conditional distributions, with the Metropolis-within-Gibbs algorithm standing out as a well-known example. Despite their widespread use in both statistical and non-statistical applications, little is known about their convergence properties. This article introduces novel methods for establishing bounds on the convergence rates of certain reversible hybrid Gibbs samplers. In particular, we examine the convergence characteristics of hybrid random-scan Gibbs algorithms. Our analysis reveals that the absolute spectral gap of a hybrid Gibbs chain can be bounded based on the absolute spectral gap of the exact Gibbs chain and the absolute spectral gaps of the Markov chains employed for conditional distribution approximations. We also provide a convergence bound of similar flavors for hybrid data augmentation algorithms, extending existing works on the topic. The general bounds are applied to three examples: a random-scan Metropolis-within-Gibbs sampler, random-scan Gibbs samplers with block updates, and a hybrid slice sampler.

翻译：混合吉布斯采样器是一类重要的近似吉布斯算法，其利用马尔可夫链来近似条件分布，其中Metropolis-within-Gibbs算法便是一个众所周知的例子。尽管它们在统计与非统计应用中广泛使用，但其收敛性质却鲜为人知。本文提出了建立某些可逆混合吉布斯采样器收敛速率界的新方法。特别地，我们研究了混合随机扫描吉布斯算法的收敛特性。我们的分析表明，混合吉布斯链的绝对谱隙可以根据精确吉布斯链的绝对谱隙以及用于条件分布近似的马尔可夫链的绝对谱隙来界定。我们还为混合数据增强算法提供了一个具有类似特征的收敛界，从而扩展了该主题的现有工作。将一般界应用于三个示例：随机扫描Metropolis-within-Gibbs采样器、采用块更新的随机扫描吉布斯采样器以及混合切片采样器。