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, very little is known about their convergence properties. This article introduces novel methods for establishing bounds on the convergence rates of hybrid Gibbs samplers. In particular, we examine the convergence characteristics of hybrid random-scan Gibbs and data augmentation algorithms. Our analysis reveals that the absolute spectral gap of a reversible hybrid 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. The new techniques are applied to four algorithms: a random-scan Metropolis-within-Gibbs sampler, a hybrid proximal 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，它利用马尔可夫链来近似条件分布，其中"吉布斯内部含Metropolis"算法是一个著名的例子。尽管在统计和非统计应用中广泛使用，但关于其收敛性质的研究却非常有限。本文提出了建立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采样器和数据增强算法的收敛特性。我们的分析表明，可逆混合链的绝对谱隙可以由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。这些新技术被应用于四种算法：随机扫描Metropolis-within-Gibbs采样器、混合近端采样器、具有块更新的随机扫描吉布斯采样器以及混合切片采样器。