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 confirms that the absolute spectral gap of a 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. For application, we study the convergence properties of four practical hybrid Gibbs 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，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范例。尽管这类算法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。本文提出了建立混合吉布斯采样器收敛率界的新方法。我们特别考察了混合随机扫描吉布斯算法与数据增强算法的收敛特性。分析证实，混合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进行界定。在应用方面，我们研究了四种实用混合吉布斯算法的收敛性质：随机扫描Metropolis-within-Gibbs采样器、混合近端采样器、带块更新的随机扫描吉布斯采样器以及混合切片采样器。