  Gibbs sampling is a widely popular Markov chain Monte Carlo algorithm that
can be used to analyze intractable posterior distributions associated with
Bayesian hierarchical models. There are two standard versions of the Gibbs
sampler: The systematic scan (SS) version, where all variables are updated at
each iteration, and the random scan (RS) version, where a single, randomly
selected variable is updated at each iteration. The literature comparing the
theoretical properties of SS and RS Gibbs samplers is reviewed, and an
alternative hybrid scan Gibbs sampler is introduced, which is particularly well
suited to Bayesian models with latent variables. The word "hybrid" reflects the
fact that the scan used within this algorithm has both systematic and random
elements. Indeed, at each iteration, one updates the entire set of latent
variables, along with a randomly chosen block of the remaining variables. The
hybrid scan (HS) Gibbs sampler has important advantages over the two standard
scan Gibbs samplers. Firstly, the HS algorithm is often easier to analyze from
a theoretical standpoint. In particular, it can be much easier to establish the
geometric ergodicity of a HS Gibbs Markov chain than to do the same for the
corresponding SS and RS versions. Secondly, the sandwich methodology developed
in Hobert and Marchev (2008), which is also reviewed, can be applied to the HS
Gibbs algorithm (but not to the standard scan Gibbs samplers). It is shown
that, under weak regularity conditions, adding sandwich steps to the HS Gibbs
sampler always results in a theoretically superior algorithm. Three specific
Bayesian hierarchical models of varying complexity are used to illustrate the
results. One is a simple location-scale model for data from the Student's $t$
distribution, which is used as a pedagogical tool. The other two are
sophisticated, yet practical Bayesian regression models.
