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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 article 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 article 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 Poisson mixture, a zero-inflated Poisson model and a spatial autoregressive combined model. The applications show that the AMIS-INLA approach, in general, outperforms the other methods compared, but the IS-INLA algorithm could be considered for faster inference when good proposals are available.  Supplementary materials  for this article are available online.

Importance Sampling with the Integrated Nested Laplace Approximation