  Double hierarchical generalized linear models (DHGLM) are a family of models
that are flexible enough as to model hierarchically the mean and scale
parameters. In a Bayesian framework, fitting highly parameterized hierarchical
models is challenging when this problem is addressed using typical Markov chain
Monte Carlo (MCMC) methods due to the potential high correlation between
different parameters and effects in the model. The integrated nested Laplace
approximation (INLA) could be considered instead to avoid dealing with these
problems. However, DHGLM do not fit within the latent Gaussian Markov random
field (GMRF) models that INLA can fit.
  In this paper we show how to fit DHGLM with INLA by combining INLA and
importance sampling (IS) algorithms. In particular, we will illustrate how to
split DHGLM into submodels that can be fitted with INLA so that the remainder
of the parameters are fit using adaptive multiple IS (AMIS) with the aid of the
graphical representation of the hierarchical model. This is illustrated using a
simulation study on three different types of models and two real data examples.
