  The Integrated Nested Laplace Approximation (INLA) is a deterministic
approach to Bayesian inference on latent Gaussian models (LGMs) and focuses on
fast and accurate approximation of posterior marginals for the parameters in
the models. Recently, methods have been developed to extend this class of
models to those that can be expressed as conditional LGMs by fixing some of the
parameters in the models to descriptive values. These methods differ in the
manner descriptive values are chosen. This paper proposes to combine importance
sampling with INLA (IS-INLA), and extends this approach with the more robust
adaptive multiple importance sampling algorithm combined with INLA (AMIS-INLA).
  This paper gives a comparison between these approaches and existing methods
on a series of applications with simulated and observed datasets and evaluates
their performance based on accuracy, efficiency, and robustness. The approaches
are validated by exact posteriors in a simple bivariate linear model; then,
they are applied to a Bayesian lasso model, a Bayesian imputation of missing
covariate values, and lastly, in parametric Bayesian quantile regression. The
applications show that the AMIS-INLA approach, in general, outperforms the
other methods, but the IS-INLA algorithm could be considered for faster
inference when good proposals are available.
