  Generalized linear mixed models (GLMMs) are often used for analyzing
correlated non-Gaussian data. The likelihood function in a GLMM is available
only as a high dimensional integral, and thus closed-form inference and
prediction are not possible for GLMMs. Since the likelihood is not available in
a closed-form, the associated posterior densities in Bayesian GLMMs are also
intractable. Generally, Markov chain Monte Carlo (MCMC) algorithms are used for
conditional simulation in GLMMs and exploring these posterior densities. In
this article, we present different state of the art MCMC algorithms for fitting
GLMMs. These MCMC algorithms include efficient data augmentation strategies, as
well as diffusions based and Hamiltonian dynamics based methods. The Langevin
and Hamiltonian Monte Carlo methods presented here are applicable to any GLMMs,
and are illustrated using three most popular GLMMs, namely, the logistic and
probit GLMMs for binomial data and the Poisson-log GLMM for count data. We also
present efficient data augmentation algorithms for probit and logistic GLMMs.
Some of these algorithms are compared using a numerical example.
