  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.
