Abstract
The key operation in Bayesian inference is to compute high-dimensional integrals. An old approximate technique is the Laplace method or approximation, which dates back to Pierre-Simon Laplace (1774). This simple idea approximates the integrand with a second-order Taylor expansion around the mode and computes the integral analytically. By developing a nested version of this classical idea, combined with modern numerical techniques for sparse matrices, we obtain the approach of integrated nested Laplace approximations (INLA) to do approximate Bayesian inference for latent Gaussian models (LGMs). LGMs represent an important model abstraction for Bayesian inference and include a large proportion of the statistical models used today. In this review, we discuss the reasons for the success of the INLA approach, the R-INLA package, why it is so accurate, why the approximations are very quick to compute, and why LGMs make such a useful concept for Bayesian computing.