ABSTRACT
The integrated Laplace approximation (INLA) methodology was first introduced by Rue et al., followed by developments in Martins et al., and is most recently reviewed in Rue et al. It is a deterministic approach to approximate Bayesian inference for latent Gaussian models (LGMs). This chapter reveals the "secrets" that make INLA successful. There are three key components required by INLA: the LGM framework, a Gaussian Markov random field (GMRF) and the Laplace approximation. The chapter introduces these components using a top-down approach, starting with LGMs and the type of statistical models that may be viewed as LGMs. It discusses the concept of a GMRF, a class of Gaussian processes that are computationally efficient within this formulation. The chapter illustrates how INLA makes use of the Laplace approximation, an old technique for approximating integrals, to perform accurate and fast Bayesian inference on LGMs. It describes an intermediate approach between full numerical integration and the plug-in method.
