  Generalized linear mixed models (GLMMs) are commonly used to
  analyze correlated discrete or continuous response data. In
  Bayesian GLMMs, the often-used improper priors may yield
  undesirable improper posterior distributions. Thus, verifying
  posterior propriety is crucial for valid applications of Bayesian
  GLMMs with improper priors. Here, we consider the popular improper
  uniform prior on the regression coefficients and several proper or
  improper priors, including the widely used gamma and power priors
  on the variance components of the random effects. We also
  construct an approximate Jeffreys' prior for objective Bayesian
  analysis of GLMMs. We derive necessary and sufficient conditions
  for posterior propriety for Bayesian GLMMs where the response
  variables have distributions from the exponential family. For the
  two most widely used GLMMs, namely, the binomial and Poisson
  GLMMs, we further refine our results by providing easily
  verifiable conditions compared to the currently available
  results. Finally, we use examples involving one-way and two-way
  random effects models to demonstrate the theoretical results
  derived here.
